Properties

Degree $4$
Conductor $5531904$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 14·5-s + 27·9-s − 18·11-s − 48·13-s − 84·15-s − 34·17-s − 16·19-s − 110·23-s + 74·25-s + 108·27-s + 212·29-s − 136·31-s − 108·33-s − 24·37-s − 288·39-s − 694·41-s + 584·43-s − 378·45-s − 316·47-s − 204·51-s + 560·53-s + 252·55-s − 96·57-s − 492·59-s + 604·61-s + 672·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.25·5-s + 9-s − 0.493·11-s − 1.02·13-s − 1.44·15-s − 0.485·17-s − 0.193·19-s − 0.997·23-s + 0.591·25-s + 0.769·27-s + 1.35·29-s − 0.787·31-s − 0.569·33-s − 0.106·37-s − 1.18·39-s − 2.64·41-s + 2.07·43-s − 1.25·45-s − 0.980·47-s − 0.560·51-s + 1.45·53-s + 0.617·55-s − 0.223·57-s − 1.08·59-s + 1.26·61-s + 1.28·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.370399596\)
\(L(\frac12)\) \(\approx\) \(3.370399596\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 14 T + 122 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 18 T + 1150 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 48 T + 4262 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 2 p T - 4222 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 16 T + 10950 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 110 T + 18686 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 212 T + 57182 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 136 T + 61374 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 24 T - 18202 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 694 T + 243914 T^{2} + 694 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 584 T + 232950 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 316 T + 231902 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 560 T + 369782 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 492 T + 351622 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 604 T + 406398 T^{2} - 604 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1020 T + 843926 T^{2} - 1020 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1710 T + 1336222 T^{2} - 1710 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1312 T + 1201998 T^{2} - 1312 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 556 T + 751134 T^{2} - 556 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 264 T + 979750 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 70 T - 186262 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 136 T + 1812270 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.575538226071418621300594096551, −8.306989345462196982494481588068, −8.058125748457096858027425184698, −7.987728412473259123450125143397, −7.20771883589076009047012964608, −7.13301480205609200915360921693, −6.64941730388337690770852370244, −6.36756352232415575144649635388, −5.42609436563531896250073384390, −5.28959162392980138117898220123, −4.68700379603590535584730063871, −4.39299688390228512666611454646, −3.72097414335759003982034198056, −3.71445667165639460204541437344, −3.11133454528342087656805183134, −2.52315834012646762498768932446, −2.15858186618317234552972511798, −1.79623628339182733818883260093, −0.66564183129699699194764421106, −0.49064354949624255040392876559, 0.49064354949624255040392876559, 0.66564183129699699194764421106, 1.79623628339182733818883260093, 2.15858186618317234552972511798, 2.52315834012646762498768932446, 3.11133454528342087656805183134, 3.71445667165639460204541437344, 3.72097414335759003982034198056, 4.39299688390228512666611454646, 4.68700379603590535584730063871, 5.28959162392980138117898220123, 5.42609436563531896250073384390, 6.36756352232415575144649635388, 6.64941730388337690770852370244, 7.13301480205609200915360921693, 7.20771883589076009047012964608, 7.987728412473259123450125143397, 8.058125748457096858027425184698, 8.306989345462196982494481588068, 8.575538226071418621300594096551

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