Properties

Label 2-1216-1.1-c3-0-53
Degree $2$
Conductor $1216$
Sign $-1$
Analytic cond. $71.7463$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 17.2·5-s − 12.3·7-s − 26·9-s + 39.2·11-s − 18.9·13-s + 17.2·15-s + 100.·17-s + 19·19-s + 12.3·21-s + 158.·23-s + 173.·25-s + 53·27-s − 141.·29-s + 215.·31-s − 39.2·33-s + 213.·35-s − 209.·37-s + 18.9·39-s + 276.·41-s − 166.·43-s + 449.·45-s − 60.5·47-s − 190.·49-s − 100.·51-s + 76.2·53-s − 679.·55-s + ⋯
L(s)  = 1  − 0.192·3-s − 1.54·5-s − 0.667·7-s − 0.962·9-s + 1.07·11-s − 0.403·13-s + 0.297·15-s + 1.42·17-s + 0.229·19-s + 0.128·21-s + 1.43·23-s + 1.39·25-s + 0.377·27-s − 0.904·29-s + 1.24·31-s − 0.207·33-s + 1.03·35-s − 0.932·37-s + 0.0777·39-s + 1.05·41-s − 0.591·43-s + 1.48·45-s − 0.187·47-s − 0.554·49-s − 0.274·51-s + 0.197·53-s − 1.66·55-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: no
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 19T \)
good3 \( 1 + T + 27T^{2} \)
5 \( 1 + 17.2T + 125T^{2} \)
7 \( 1 + 12.3T + 343T^{2} \)
11 \( 1 - 39.2T + 1.33e3T^{2} \)
13 \( 1 + 18.9T + 2.19e3T^{2} \)
17 \( 1 - 100.T + 4.91e3T^{2} \)
23 \( 1 - 158.T + 1.21e4T^{2} \)
29 \( 1 + 141.T + 2.43e4T^{2} \)
31 \( 1 - 215.T + 2.97e4T^{2} \)
37 \( 1 + 209.T + 5.06e4T^{2} \)
41 \( 1 - 276.T + 6.89e4T^{2} \)
43 \( 1 + 166.T + 7.95e4T^{2} \)
47 \( 1 + 60.5T + 1.03e5T^{2} \)
53 \( 1 - 76.2T + 1.48e5T^{2} \)
59 \( 1 + 575.T + 2.05e5T^{2} \)
61 \( 1 - 91.0T + 2.26e5T^{2} \)
67 \( 1 - 444.T + 3.00e5T^{2} \)
71 \( 1 + 943.T + 3.57e5T^{2} \)
73 \( 1 + 569.T + 3.89e5T^{2} \)
79 \( 1 - 336.T + 4.93e5T^{2} \)
83 \( 1 + 603.T + 5.71e5T^{2} \)
89 \( 1 - 1.63e3T + 7.04e5T^{2} \)
97 \( 1 - 319.T + 9.12e5T^{2} \)
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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

−8.917893205469915773158969755520, −8.083738722835255727420789698462, −7.33711514859523866606934098749, −6.56410156099193031788154077670, −5.55413727175344320027893379523, −4.55136195158787050176654012662, −3.49115604823327394888079237802, −3.02263423017133149104774076181, −1.04993588526087211607630818539, 0, 1.04993588526087211607630818539, 3.02263423017133149104774076181, 3.49115604823327394888079237802, 4.55136195158787050176654012662, 5.55413727175344320027893379523, 6.56410156099193031788154077670, 7.33711514859523866606934098749, 8.083738722835255727420789698462, 8.917893205469915773158969755520

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