Properties

Label 4-294e2-1.1-c3e2-0-8
Degree $4$
Conductor $86436$
Sign $1$
Analytic cond. $300.903$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 6·3-s + 12·4-s + 12·5-s + 24·6-s − 32·8-s + 27·9-s − 48·10-s − 4·11-s − 72·12-s + 48·13-s − 72·15-s + 80·16-s + 84·17-s − 108·18-s − 72·19-s + 144·20-s + 16·22-s − 308·23-s + 192·24-s − 44·25-s − 192·26-s − 108·27-s − 80·29-s + 288·30-s + 384·31-s − 192·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.07·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.51·10-s − 0.109·11-s − 1.73·12-s + 1.02·13-s − 1.23·15-s + 5/4·16-s + 1.19·17-s − 1.41·18-s − 0.869·19-s + 1.60·20-s + 0.155·22-s − 2.79·23-s + 1.63·24-s − 0.351·25-s − 1.44·26-s − 0.769·27-s − 0.512·29-s + 1.75·30-s + 2.22·31-s − 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86436\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(300.903\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{294} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86436,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.225925987\)
\(L(\frac12)\) \(\approx\) \(1.225925987\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
3$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 12 T + 188 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 4 T - 862 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 48 T + 4088 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 84 T + 676 p T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 72 T + 14622 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 308 T + 44522 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 80 T + 18626 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 384 T + 92918 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 536 T + 159018 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 756 T + 278276 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 400 T + 142566 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 312 T + 222182 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 52 T + 241982 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 864 T + 596990 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 1416 T + 938664 T^{2} - 1416 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 144 T + 550262 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1524 T + 1208266 T^{2} + 1524 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 744 T + 241296 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 976 T + 532734 T^{2} - 976 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 312 T - 644698 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 108 T + 1330436 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 744 T + 432480 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.32060650464516985981258843415, −11.05852364847807429747357750622, −10.54657226267419943469028963146, −9.947765621352049508084289630428, −9.817954818624224272394093429000, −9.597596540950967412951135462825, −8.709411950258291662473298027986, −8.217287686355713759507596435813, −7.78218209660205036588708099713, −7.39188270965617798545885839387, −6.39613473213591813443088913526, −6.17403852664435487822821993804, −5.84128024223951553870130717180, −5.53319553785407777438267661384, −4.11264196605981104760747501672, −4.08187349126039454931043441182, −2.47008666819387540150876888750, −2.20810816843075961671558928849, −1.07004601042491447273294594266, −0.69779907337986548598530769885, 0.69779907337986548598530769885, 1.07004601042491447273294594266, 2.20810816843075961671558928849, 2.47008666819387540150876888750, 4.08187349126039454931043441182, 4.11264196605981104760747501672, 5.53319553785407777438267661384, 5.84128024223951553870130717180, 6.17403852664435487822821993804, 6.39613473213591813443088913526, 7.39188270965617798545885839387, 7.78218209660205036588708099713, 8.217287686355713759507596435813, 8.709411950258291662473298027986, 9.597596540950967412951135462825, 9.817954818624224272394093429000, 9.947765621352049508084289630428, 10.54657226267419943469028963146, 11.05852364847807429747357750622, 11.32060650464516985981258843415

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