Properties

Label 2-1216-8.5-c1-0-29
Degree $2$
Conductor $1216$
Sign $-0.707 + 0.707i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 4i·5-s − 7-s − 6·9-s + 5i·13-s + 12·15-s − 5·17-s + i·19-s − 3i·21-s + 3·23-s − 11·25-s − 9i·27-s − 7i·29-s − 10·31-s + 4i·35-s + ⋯
L(s)  = 1  + 1.73i·3-s − 1.78i·5-s − 0.377·7-s − 2·9-s + 1.38i·13-s + 3.09·15-s − 1.21·17-s + 0.229i·19-s − 0.654i·21-s + 0.625·23-s − 2.20·25-s − 1.73i·27-s − 1.29i·29-s − 1.79·31-s + 0.676i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.707 + 0.707i)\)

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$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - iT \)
good3 \( 1 - 3iT - 3T^{2} \)
5 \( 1 + 4iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + 7iT - 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 - iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.560423801687851858035516854593, −8.789738150830262684436493570506, −8.364989284097332836323452357892, −6.81893578599937410538296245717, −5.64627389990694439814382002943, −4.94020677998479479840763424810, −4.28806962137891571388993652914, −3.68374437429460524000993398476, −1.97657079152034106338366274028, 0, 1.77849389118508217197657670653, 2.83619747990884525794359393850, 3.34014982079073240885196077070, 5.31229610751105295105052134212, 6.33333972766039216673932136650, 6.75490869545100678888110266108, 7.37759537552068899213166869092, 8.054932625377401236969486030192, 9.078914580800521084834807858492

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