Properties

Label 2-1216-1.1-c3-0-56
Degree $2$
Conductor $1216$
Sign $1$
Analytic cond. $71.7463$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + 12·5-s + 11·7-s − 2·9-s + 54·11-s − 11·13-s + 60·15-s − 93·17-s − 19·19-s + 55·21-s + 183·23-s + 19·25-s − 145·27-s + 249·29-s + 56·31-s + 270·33-s + 132·35-s + 250·37-s − 55·39-s + 240·41-s + 196·43-s − 24·45-s − 168·47-s − 222·49-s − 465·51-s − 435·53-s + 648·55-s + ⋯
L(s)  = 1  + 0.962·3-s + 1.07·5-s + 0.593·7-s − 0.0740·9-s + 1.48·11-s − 0.234·13-s + 1.03·15-s − 1.32·17-s − 0.229·19-s + 0.571·21-s + 1.65·23-s + 0.151·25-s − 1.03·27-s + 1.59·29-s + 0.324·31-s + 1.42·33-s + 0.637·35-s + 1.11·37-s − 0.225·39-s + 0.914·41-s + 0.695·43-s − 0.0795·45-s − 0.521·47-s − 0.647·49-s − 1.27·51-s − 1.12·53-s + 1.58·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $1$
Analytic conductor: \(71.7463\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.399536052\)
\(L(\frac12)\) \(\approx\) \(4.399536052\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + p T \)
good3 \( 1 - 5 T + p^{3} T^{2} \)
5 \( 1 - 12 T + p^{3} T^{2} \)
7 \( 1 - 11 T + p^{3} T^{2} \)
11 \( 1 - 54 T + p^{3} T^{2} \)
13 \( 1 + 11 T + p^{3} T^{2} \)
17 \( 1 + 93 T + p^{3} T^{2} \)
23 \( 1 - 183 T + p^{3} T^{2} \)
29 \( 1 - 249 T + p^{3} T^{2} \)
31 \( 1 - 56 T + p^{3} T^{2} \)
37 \( 1 - 250 T + p^{3} T^{2} \)
41 \( 1 - 240 T + p^{3} T^{2} \)
43 \( 1 - 196 T + p^{3} T^{2} \)
47 \( 1 + 168 T + p^{3} T^{2} \)
53 \( 1 + 435 T + p^{3} T^{2} \)
59 \( 1 + 195 T + p^{3} T^{2} \)
61 \( 1 - 358 T + p^{3} T^{2} \)
67 \( 1 - 961 T + p^{3} T^{2} \)
71 \( 1 + 246 T + p^{3} T^{2} \)
73 \( 1 - 353 T + p^{3} T^{2} \)
79 \( 1 + 34 T + p^{3} T^{2} \)
83 \( 1 + 234 T + p^{3} T^{2} \)
89 \( 1 + 168 T + p^{3} T^{2} \)
97 \( 1 - 758 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.212837792718651816403263863502, −8.763413412787610939658558463615, −7.917838037139970653434127218980, −6.74142297107563926188412553848, −6.21799390374130640479761027434, −4.98591942561872054259420218180, −4.15369623708891343604603768406, −2.89145229083996227486700512369, −2.12938759684838702925342999400, −1.10575574965442025606339178560, 1.10575574965442025606339178560, 2.12938759684838702925342999400, 2.89145229083996227486700512369, 4.15369623708891343604603768406, 4.98591942561872054259420218180, 6.21799390374130640479761027434, 6.74142297107563926188412553848, 7.917838037139970653434127218980, 8.763413412787610939658558463615, 9.212837792718651816403263863502

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