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Properties

Label 8-2310e4-1.1-c3e4-0-2
Degree $8$
Conductor $2.847\times 10^{13}$
Sign $1$
Analytic cond. $3.45073\times 10^{8}$
Root an. cond. $11.6745$
Motivic weight $3$
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  − 8·2-s + 12·3-s + 40·4-s − 20·5-s − 96·6-s − 28·7-s − 160·8-s + 90·9-s + 160·10-s + 44·11-s + 480·12-s + 84·13-s + 224·14-s − 240·15-s + 560·16-s + 57·17-s − 720·18-s − 17·19-s − 800·20-s − 336·21-s − 352·22-s + 23-s − 1.92e3·24-s + 250·25-s − 672·26-s + 540·27-s − 1.12e3·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s − 6.53·6-s − 1.51·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s + 1.20·11-s + 11.5·12-s + 1.79·13-s + 4.27·14-s − 4.13·15-s + 35/4·16-s + 0.813·17-s − 9.42·18-s − 0.205·19-s − 8.94·20-s − 3.49·21-s − 3.41·22-s + 0.00906·23-s − 16.3·24-s + 2·25-s − 5.06·26-s + 3.84·27-s − 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3.45073\times 10^{8}\)
Root analytic conductor: \(11.6745\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(6.352600859\)
\(L(\frac12)\) \(\approx\) \(6.352600859\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
3$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
7$C_1$ \( ( 1 + p T )^{4} \)
11$C_1$ \( ( 1 - p T )^{4} \)
good13$C_2 \wr S_4$ \( 1 - 84 T + 7200 T^{2} - 416252 T^{3} + 24707662 T^{4} - 416252 p^{3} T^{5} + 7200 p^{6} T^{6} - 84 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 57 T + 15904 T^{2} - 845887 T^{3} + 108343086 T^{4} - 845887 p^{3} T^{5} + 15904 p^{6} T^{6} - 57 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 17 T + 6448 T^{2} - 562991 T^{3} + 12947502 T^{4} - 562991 p^{3} T^{5} + 6448 p^{6} T^{6} + 17 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - T + 26594 T^{2} - 1617533 T^{3} + 323285178 T^{4} - 1617533 p^{3} T^{5} + 26594 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 71 T + 18898 T^{2} + 4083907 T^{3} - 495537382 T^{4} + 4083907 p^{3} T^{5} + 18898 p^{6} T^{6} - 71 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 26 T + 42028 T^{2} + 10857426 T^{3} + 565304358 T^{4} + 10857426 p^{3} T^{5} + 42028 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 236 T + 138072 T^{2} - 27858996 T^{3} + 9369936990 T^{4} - 27858996 p^{3} T^{5} + 138072 p^{6} T^{6} - 236 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 260 T + 121300 T^{2} + 6668644 T^{3} + 1953807670 T^{4} + 6668644 p^{3} T^{5} + 121300 p^{6} T^{6} - 260 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 561 T + 183356 T^{2} + 66118841 T^{3} + 23363528086 T^{4} + 66118841 p^{3} T^{5} + 183356 p^{6} T^{6} + 561 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 6 T + 174340 T^{2} - 25161422 T^{3} + 15626667702 T^{4} - 25161422 p^{3} T^{5} + 174340 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 1133 T + 938290 T^{2} - 536198087 T^{3} + 234902819066 T^{4} - 536198087 p^{3} T^{5} + 938290 p^{6} T^{6} - 1133 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 521 T + 459640 T^{2} - 77832833 T^{3} + 70376324606 T^{4} - 77832833 p^{3} T^{5} + 459640 p^{6} T^{6} - 521 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 1449 T + 1204278 T^{2} + 763000835 T^{3} + 403710550034 T^{4} + 763000835 p^{3} T^{5} + 1204278 p^{6} T^{6} + 1449 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 868 T + 488944 T^{2} - 47366876 T^{3} - 74846809682 T^{4} - 47366876 p^{3} T^{5} + 488944 p^{6} T^{6} + 868 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 1158 T + 1577876 T^{2} + 1169832158 T^{3} + 869095375990 T^{4} + 1169832158 p^{3} T^{5} + 1577876 p^{6} T^{6} + 1158 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 222 T + 844768 T^{2} + 397547314 T^{3} + 368112223774 T^{4} + 397547314 p^{3} T^{5} + 844768 p^{6} T^{6} + 222 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 820 T + 1568508 T^{2} + 767093668 T^{3} + 976962578502 T^{4} + 767093668 p^{3} T^{5} + 1568508 p^{6} T^{6} + 820 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 333 T + 1797412 T^{2} - 492781789 T^{3} + 1464556239014 T^{4} - 492781789 p^{3} T^{5} + 1797412 p^{6} T^{6} - 333 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 561 T + 623656 T^{2} + 280822711 T^{3} + 731971769006 T^{4} + 280822711 p^{3} T^{5} + 623656 p^{6} T^{6} + 561 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 799 T + 3788050 T^{2} - 2179256481 T^{3} + 5247416930842 T^{4} - 2179256481 p^{3} T^{5} + 3788050 p^{6} T^{6} - 799 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.37669071149938056233612981697, −5.83410221463650598242873516762, −5.66251089780151088843698684126, −5.66199914832082626806922683255, −5.63446348768997390614439031258, −4.53823689072796284110229962556, −4.50027529905961398144421717733, −4.41173449005842746585498668263, −4.39654444214048894674912590101, −3.72280582020852031915538339538, −3.60447023474376980542578152859, −3.59196984158740848515568746359, −3.35841666876342902996055994539, −3.05556465277221353601262558749, −2.84806693669743129858006748401, −2.75883295911045725049670939395, −2.62573677008805812490580124844, −1.86188573231911759230390539161, −1.75799700988165479990279947939, −1.62219919153571287318116034160, −1.51550856702077916404860568339, −0.792595796727128410846748606457, −0.70214378772457254918890476105, −0.56076532903460013215501091060, −0.48076666841178433827306859271, 0.48076666841178433827306859271, 0.56076532903460013215501091060, 0.70214378772457254918890476105, 0.792595796727128410846748606457, 1.51550856702077916404860568339, 1.62219919153571287318116034160, 1.75799700988165479990279947939, 1.86188573231911759230390539161, 2.62573677008805812490580124844, 2.75883295911045725049670939395, 2.84806693669743129858006748401, 3.05556465277221353601262558749, 3.35841666876342902996055994539, 3.59196984158740848515568746359, 3.60447023474376980542578152859, 3.72280582020852031915538339538, 4.39654444214048894674912590101, 4.41173449005842746585498668263, 4.50027529905961398144421717733, 4.53823689072796284110229962556, 5.63446348768997390614439031258, 5.66199914832082626806922683255, 5.66251089780151088843698684126, 5.83410221463650598242873516762, 6.37669071149938056233612981697

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