Properties

Degree $4$
Conductor $49787136$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 4·11-s + 8·13-s − 4·17-s − 4·23-s + 4·25-s + 8·29-s + 8·31-s − 8·37-s − 4·41-s + 4·53-s + 16·55-s − 8·59-s + 16·61-s − 32·65-s − 4·71-s + 8·73-s − 16·79-s − 8·83-s + 16·85-s + 20·89-s + 8·97-s − 20·101-s + 8·103-s − 12·107-s − 12·113-s + 16·115-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.20·11-s + 2.21·13-s − 0.970·17-s − 0.834·23-s + 4/5·25-s + 1.48·29-s + 1.43·31-s − 1.31·37-s − 0.624·41-s + 0.549·53-s + 2.15·55-s − 1.04·59-s + 2.04·61-s − 3.96·65-s − 0.474·71-s + 0.936·73-s − 1.80·79-s − 0.878·83-s + 1.73·85-s + 2.11·89-s + 0.812·97-s − 1.99·101-s + 0.788·103-s − 1.16·107-s − 1.12·113-s + 1.49·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 168 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 64 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 208 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.78448862739273330815218123613, −7.61818738353643945942363507302, −6.96829823455528476656538533109, −6.85103921159711484048535080032, −6.31669332329476244176907301385, −6.20135593320807817315254874086, −5.60305092506540907928158259438, −5.33075773874890257026782941947, −4.73314714532105852460388231046, −4.51485706815830284709307940531, −4.13534721591455643520405491284, −3.70208936368141272254606387191, −3.50482297221063585092269416635, −3.12356535290121300001281251631, −2.37022443881284216851369091798, −2.32631613539042043642637909539, −1.26828828996516756953010643492, −1.08560958480826197450945713323, 0, 0, 1.08560958480826197450945713323, 1.26828828996516756953010643492, 2.32631613539042043642637909539, 2.37022443881284216851369091798, 3.12356535290121300001281251631, 3.50482297221063585092269416635, 3.70208936368141272254606387191, 4.13534721591455643520405491284, 4.51485706815830284709307940531, 4.73314714532105852460388231046, 5.33075773874890257026782941947, 5.60305092506540907928158259438, 6.20135593320807817315254874086, 6.31669332329476244176907301385, 6.85103921159711484048535080032, 6.96829823455528476656538533109, 7.61818738353643945942363507302, 7.78448862739273330815218123613

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