Dirichlet series
L(s) = 1 | + 32·2-s + 14·3-s + 256·4-s + 2.73e3·5-s + 448·6-s − 8.19e3·8-s − 1.82e4·9-s + 8.73e4·10-s − 4.49e4·11-s + 3.58e3·12-s + 2.00e5·13-s + 3.82e4·15-s − 2.62e5·16-s + 8.70e5·17-s − 5.82e5·18-s − 5.08e5·19-s + 6.98e5·20-s − 1.43e6·22-s − 7.98e4·23-s − 1.14e5·24-s + 4.75e6·25-s + 6.41e6·26-s − 1.31e6·27-s + 4.01e6·29-s + 1.22e6·30-s − 2.18e6·31-s − 2.09e6·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.0997·3-s + 1/2·4-s + 1.95·5-s + 0.141·6-s − 0.707·8-s − 0.925·9-s + 2.76·10-s − 0.925·11-s + 0.0498·12-s + 1.94·13-s + 0.194·15-s − 16-s + 2.52·17-s − 1.30·18-s − 0.895·19-s + 0.976·20-s − 1.30·22-s − 0.0594·23-s − 0.0705·24-s + 2.43·25-s + 2.75·26-s − 0.477·27-s + 1.05·29-s + 0.275·30-s − 0.425·31-s − 0.353·32-s + ⋯ |
Functional equation
Invariants
Degree: | \(8\) |
Conductor: | \(92236816\) = \(2^{4} \cdot 7^{8}\) |
Sign: | $1$ |
Analytic conductor: | \(6.49014\times 10^{6}\) |
Root analytic conductor: | \(7.10447\) |
Motivic weight: | \(9\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((8,\ 92236816,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\) |
Particular Values
\(L(5)\) | \(\approx\) | \(5.971012845\) |
\(L(\frac12)\) | \(\approx\) | \(5.971012845\) |
\(L(\frac{11}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 2 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )^{2} \) |
7 | \( 1 \) | ||
good | 3 | $D_4\times C_2$ | \( 1 - 14 T + 18406 T^{2} + 268688 p T^{3} - 6630473 p^{2} T^{4} + 268688 p^{10} T^{5} + 18406 p^{18} T^{6} - 14 p^{27} T^{7} + p^{36} T^{8} \) |
5 | $D_4\times C_2$ | \( 1 - 546 p T + 539986 p T^{2} - 92461824 p^{2} T^{3} + 168783249711 p^{2} T^{4} - 92461824 p^{11} T^{5} + 539986 p^{19} T^{6} - 546 p^{28} T^{7} + p^{36} T^{8} \) | |
11 | $D_4\times C_2$ | \( 1 + 44940 T - 659930182 T^{2} - 91514090304000 T^{3} - 3142135669968295557 T^{4} - 91514090304000 p^{9} T^{5} - 659930182 p^{18} T^{6} + 44940 p^{27} T^{7} + p^{36} T^{8} \) | |
13 | $D_{4}$ | \( ( 1 - 7714 p T + 23606428002 T^{2} - 7714 p^{10} T^{3} + p^{18} T^{4} )^{2} \) | |
17 | $D_4\times C_2$ | \( 1 - 870408 T + 336859082354 T^{2} - 159785367173375328 T^{3} + \)\(73\!\cdots\!83\)\( T^{4} - 159785367173375328 p^{9} T^{5} + 336859082354 p^{18} T^{6} - 870408 p^{27} T^{7} + p^{36} T^{8} \) | |
19 | $D_4\times C_2$ | \( 1 + 508774 T - 33946815626 T^{2} - 9441174434884976 p T^{3} - \)\(28\!\cdots\!81\)\( p^{2} T^{4} - 9441174434884976 p^{10} T^{5} - 33946815626 p^{18} T^{6} + 508774 p^{27} T^{7} + p^{36} T^{8} \) | |
23 | $D_4\times C_2$ | \( 1 + 79800 T - 151305274562 p T^{2} - 9250094246400000 T^{3} + \)\(88\!\cdots\!07\)\( T^{4} - 9250094246400000 p^{9} T^{5} - 151305274562 p^{19} T^{6} + 79800 p^{27} T^{7} + p^{36} T^{8} \) | |
29 | $D_{4}$ | \( ( 1 - 2006328 T - 2889837567866 T^{2} - 2006328 p^{9} T^{3} + p^{18} T^{4} )^{2} \) | |
31 | $D_4\times C_2$ | \( 1 + 2188732 T - 45720846780974 T^{2} - 5182588568359774208 T^{3} + \)\(17\!\cdots\!59\)\( T^{4} - 5182588568359774208 p^{9} T^{5} - 45720846780974 p^{18} T^{6} + 2188732 p^{27} T^{7} + p^{36} T^{8} \) | |
37 | $D_4\times C_2$ | \( 1 - 20723576 T + 109318009611178 T^{2} - \)\(12\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!44\)\( p^{9} T^{5} + 109318009611178 p^{18} T^{6} - 20723576 p^{27} T^{7} + p^{36} T^{8} \) | |
41 | $D_{4}$ | \( ( 1 - 19016592 T + 126947391521038 T^{2} - 19016592 p^{9} T^{3} + p^{18} T^{4} )^{2} \) | |
43 | $D_{4}$ | \( ( 1 - 4193716 T + 733843976191350 T^{2} - 4193716 p^{9} T^{3} + p^{18} T^{4} )^{2} \) | |
47 | $D_4\times C_2$ | \( 1 - 74542524 T + 2021776853206898 T^{2} - \)\(96\!\cdots\!56\)\( T^{3} + \)\(50\!\cdots\!03\)\( T^{4} - \)\(96\!\cdots\!56\)\( p^{9} T^{5} + 2021776853206898 p^{18} T^{6} - 74542524 p^{27} T^{7} + p^{36} T^{8} \) | |
53 | $D_4\times C_2$ | \( 1 - 3239748 T - 6094285004204638 T^{2} + \)\(16\!\cdots\!52\)\( T^{3} + \)\(26\!\cdots\!43\)\( T^{4} + \)\(16\!\cdots\!52\)\( p^{9} T^{5} - 6094285004204638 p^{18} T^{6} - 3239748 p^{27} T^{7} + p^{36} T^{8} \) | |
59 | $D_4\times C_2$ | \( 1 - 133642362 T - 1040757032736970 T^{2} - \)\(21\!\cdots\!32\)\( T^{3} + \)\(12\!\cdots\!59\)\( T^{4} - \)\(21\!\cdots\!32\)\( p^{9} T^{5} - 1040757032736970 p^{18} T^{6} - 133642362 p^{27} T^{7} + p^{36} T^{8} \) | |
61 | $D_4\times C_2$ | \( 1 + 227801686 T + 17261381012831290 T^{2} + \)\(25\!\cdots\!64\)\( T^{3} + \)\(45\!\cdots\!19\)\( T^{4} + \)\(25\!\cdots\!64\)\( p^{9} T^{5} + 17261381012831290 p^{18} T^{6} + 227801686 p^{27} T^{7} + p^{36} T^{8} \) | |
67 | $D_4\times C_2$ | \( 1 + 332930272 T + 35027578907947594 T^{2} + \)\(71\!\cdots\!12\)\( T^{3} + \)\(19\!\cdots\!43\)\( T^{4} + \)\(71\!\cdots\!12\)\( p^{9} T^{5} + 35027578907947594 p^{18} T^{6} + 332930272 p^{27} T^{7} + p^{36} T^{8} \) | |
71 | $D_{4}$ | \( ( 1 + 167985720 T + 6569741497979662 T^{2} + 167985720 p^{9} T^{3} + p^{18} T^{4} )^{2} \) | |
73 | $D_4\times C_2$ | \( 1 - 44684276 T - 32068117882924694 T^{2} + \)\(37\!\cdots\!56\)\( T^{3} - \)\(24\!\cdots\!57\)\( T^{4} + \)\(37\!\cdots\!56\)\( p^{9} T^{5} - 32068117882924694 p^{18} T^{6} - 44684276 p^{27} T^{7} + p^{36} T^{8} \) | |
79 | $D_4\times C_2$ | \( 1 + 269642776 T - 173875969301015006 T^{2} + \)\(18\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!19\)\( T^{4} + \)\(18\!\cdots\!44\)\( p^{9} T^{5} - 173875969301015006 p^{18} T^{6} + 269642776 p^{27} T^{7} + p^{36} T^{8} \) | |
83 | $D_{4}$ | \( ( 1 + 183105762 T + 297182791992067342 T^{2} + 183105762 p^{9} T^{3} + p^{18} T^{4} )^{2} \) | |
89 | $D_4\times C_2$ | \( 1 + 791657748 T - 124556565155958790 T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(22\!\cdots\!19\)\( T^{4} + \)\(40\!\cdots\!48\)\( p^{9} T^{5} - 124556565155958790 p^{18} T^{6} + 791657748 p^{27} T^{7} + p^{36} T^{8} \) | |
97 | $D_{4}$ | \( ( 1 + 4169480 T + 1069351625837487534 T^{2} + 4169480 p^{9} T^{3} + p^{18} T^{4} )^{2} \) | |
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Imaginary part of the first few zeros on the critical line
−8.457821077787632450779253738310, −8.359816423400239563175997564292, −7.59185549490396757351729469747, −7.45143070914079972536975733702, −7.39451175187236986955803582854, −6.62650932307182861552935901467, −6.19138791357380286484549295909, −5.94451188054806254530605775549, −5.85758145142843955379175893565, −5.75936436071598418968147758164, −5.67050177384797059621280639570, −4.97953088996962982127101229139, −4.73110499567051562500416747843, −4.42609127690259961404881499688, −3.98086046828716066354719322659, −3.61652503339674133477473093915, −3.32233848006687623505782320095, −2.83249417001157879721520183527, −2.58262961682268735291212805468, −2.51935301587474608076018529187, −1.95536200967152296922828252949, −1.15494922149524804516101931564, −1.14693827213490772206153007722, −1.02460189842318983258086508974, −0.17409113828656555799705887972, 0.17409113828656555799705887972, 1.02460189842318983258086508974, 1.14693827213490772206153007722, 1.15494922149524804516101931564, 1.95536200967152296922828252949, 2.51935301587474608076018529187, 2.58262961682268735291212805468, 2.83249417001157879721520183527, 3.32233848006687623505782320095, 3.61652503339674133477473093915, 3.98086046828716066354719322659, 4.42609127690259961404881499688, 4.73110499567051562500416747843, 4.97953088996962982127101229139, 5.67050177384797059621280639570, 5.75936436071598418968147758164, 5.85758145142843955379175893565, 5.94451188054806254530605775549, 6.19138791357380286484549295909, 6.62650932307182861552935901467, 7.39451175187236986955803582854, 7.45143070914079972536975733702, 7.59185549490396757351729469747, 8.359816423400239563175997564292, 8.457821077787632450779253738310