Properties

Label 8-48e4-1.1-c12e4-0-0
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $3.70457\times 10^{6}$
Root an. cond. $6.62357$
Motivic weight $12$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.04e4·5-s − 3.54e5·9-s − 1.19e7·13-s − 3.14e7·17-s − 1.44e8·25-s − 1.07e9·29-s − 6.55e9·37-s − 5.60e8·41-s − 1.07e10·45-s − 7.34e9·49-s − 1.53e11·53-s − 2.13e11·61-s − 3.63e11·65-s − 1.11e11·73-s + 9.41e10·81-s − 9.56e11·85-s + 1.07e12·89-s + 2.40e12·97-s + 2.58e12·101-s + 8.00e12·109-s + 3.86e12·113-s + 4.22e12·117-s + 9.48e12·121-s − 1.68e13·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.94·5-s − 2/3·9-s − 2.47·13-s − 1.30·17-s − 0.593·25-s − 1.81·29-s − 2.55·37-s − 0.117·41-s − 1.29·45-s − 0.530·49-s − 6.93·53-s − 4.13·61-s − 4.81·65-s − 0.738·73-s + 1/3·81-s − 2.53·85-s + 2.15·89-s + 2.88·97-s + 2.43·101-s + 4.77·109-s + 1.85·113-s + 1.64·117-s + 3.02·121-s − 4.41·125-s − 3.53·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3.70457\times 10^{6}\)
Root analytic conductor: \(6.62357\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :6, 6, 6, 6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.2013670026\)
\(L(\frac12)\) \(\approx\) \(0.2013670026\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( ( 1 + p^{11} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 - 15228 T + 84049582 p T^{2} - 15228 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 + 7346847836 T^{2} + 6254212326873420198 p^{2} T^{4} + 7346847836 p^{24} T^{6} + p^{48} T^{8} \)
11$D_4\times C_2$ \( 1 - 9486874346980 T^{2} + \)\(34\!\cdots\!42\)\( p^{2} T^{4} - 9486874346980 p^{24} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 + 5965324 T + 46306580956806 T^{2} + 5965324 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 923724 p T + 1063709195596454 T^{2} + 923724 p^{13} T^{3} + p^{24} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 5337794883427684 T^{2} + \)\(14\!\cdots\!62\)\( T^{4} - 5337794883427684 p^{24} T^{6} + p^{48} T^{8} \)
23$D_4\times C_2$ \( 1 - 58365847802924932 T^{2} + \)\(16\!\cdots\!62\)\( T^{4} - 58365847802924932 p^{24} T^{6} + p^{48} T^{8} \)
29$D_{4}$ \( ( 1 + 539788212 T + 88800945374497862 T^{2} + 539788212 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1347647351185790500 T^{2} + \)\(17\!\cdots\!22\)\( p^{2} T^{4} - 1347647351185790500 p^{24} T^{6} + p^{48} T^{8} \)
37$D_{4}$ \( ( 1 + 3278785628 T + 13999983718271264742 T^{2} + 3278785628 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 280014732 T + 5585539703810648294 T^{2} + 280014732 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!24\)\( T^{2} + \)\(89\!\cdots\!62\)\( T^{4} - \)\(15\!\cdots\!24\)\( p^{24} T^{6} + p^{48} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(23\!\cdots\!84\)\( T^{2} + \)\(40\!\cdots\!82\)\( T^{4} - \)\(23\!\cdots\!84\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_{4}$ \( ( 1 + 76800611700 T + \)\(24\!\cdots\!86\)\( T^{2} + 76800611700 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(41\!\cdots\!40\)\( T^{2} + \)\(84\!\cdots\!22\)\( T^{4} - \)\(41\!\cdots\!40\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 + 106630675004 T + \)\(60\!\cdots\!62\)\( T^{2} + 106630675004 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(12\!\cdots\!64\)\( T^{2} + \)\(16\!\cdots\!62\)\( T^{4} - \)\(12\!\cdots\!64\)\( p^{24} T^{6} + p^{48} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(49\!\cdots\!44\)\( T^{2} + \)\(10\!\cdots\!02\)\( T^{4} - \)\(49\!\cdots\!44\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 + 55901255740 T + \)\(45\!\cdots\!06\)\( T^{2} + 55901255740 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!36\)\( T^{2} + \)\(64\!\cdots\!90\)\( T^{4} - \)\(11\!\cdots\!36\)\( p^{24} T^{6} + p^{48} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(37\!\cdots\!28\)\( T^{2} + \)\(57\!\cdots\!38\)\( T^{4} - \)\(37\!\cdots\!28\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_{4}$ \( ( 1 - 535708194372 T + \)\(31\!\cdots\!38\)\( T^{2} - 535708194372 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 1203092069060 T + \)\(16\!\cdots\!78\)\( T^{2} - 1203092069060 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.063853254020563789973027597315, −9.021219220244604710933078447631, −8.500882267473063410674149185963, −7.83096676201728096760667440491, −7.59367687033610472872257076518, −7.40906428299544783335475080230, −7.29051792893444114284977826901, −6.31092791453139658797370155542, −6.21697550799940288744833372580, −6.07730146293102633183222896634, −5.98986356829620390441585189385, −5.04646124612852153313856343833, −4.91466613129498740447810505233, −4.84031779488292352249138416481, −4.55199172520043727870809179058, −3.49515901572137695565350108489, −3.40444017419728379256989355645, −3.12314549056280935497904554851, −2.40973104943003839279664903187, −2.03214083215502080288934848098, −1.94733128160114206860946372316, −1.77000019847543277706397628182, −1.32002112089258793910580658107, −0.17476924500806456276422315747, −0.17414475179433977086645762664, 0.17414475179433977086645762664, 0.17476924500806456276422315747, 1.32002112089258793910580658107, 1.77000019847543277706397628182, 1.94733128160114206860946372316, 2.03214083215502080288934848098, 2.40973104943003839279664903187, 3.12314549056280935497904554851, 3.40444017419728379256989355645, 3.49515901572137695565350108489, 4.55199172520043727870809179058, 4.84031779488292352249138416481, 4.91466613129498740447810505233, 5.04646124612852153313856343833, 5.98986356829620390441585189385, 6.07730146293102633183222896634, 6.21697550799940288744833372580, 6.31092791453139658797370155542, 7.29051792893444114284977826901, 7.40906428299544783335475080230, 7.59367687033610472872257076518, 7.83096676201728096760667440491, 8.500882267473063410674149185963, 9.021219220244604710933078447631, 9.063853254020563789973027597315

Graph of the $Z$-function along the critical line