Properties

Label 2-1216-152.75-c1-0-7
Degree $2$
Conductor $1216$
Sign $-0.924 + 0.380i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.64i·3-s + 3.46i·5-s + i·7-s − 4.00·9-s + 2.64·13-s − 9.16·15-s − 3·17-s + (−3.46 + 2.64i)19-s − 2.64·21-s + 3i·23-s − 6.99·25-s − 2.64i·27-s + 7.93·29-s − 3.46·35-s + 10.5·37-s + ⋯
L(s)  = 1  + 1.52i·3-s + 1.54i·5-s + 0.377i·7-s − 1.33·9-s + 0.733·13-s − 2.36·15-s − 0.727·17-s + (−0.794 + 0.606i)19-s − 0.577·21-s + 0.625i·23-s − 1.39·25-s − 0.509i·27-s + 1.47·29-s − 0.585·35-s + 1.73·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.924 + 0.380i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.924 + 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.376226655\)
\(L(\frac12)\) \(\approx\) \(1.376226655\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.46 - 2.64i)T \)
good3 \( 1 - 2.64iT - 3T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + 9.16iT - 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 - 7.93iT - 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 - 9.16T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + 9.16T + 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 - 9.16iT - 89T^{2} \)
97 \( 1 + 9.16iT - 97T^{2} \)
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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

−10.29751272374416617987870902488, −9.562367964363615099913669887545, −8.750258233724518417390719603640, −7.86547163225915508941138638341, −6.61832120019357949101379547979, −6.09821268832581471594804137233, −4.97436670550036931936332119937, −3.96724887507630970068491393738, −3.30990611421250955451208655633, −2.31111772095551403053240269003, 0.61462210392373565737251189020, 1.42459055477061430825202347179, 2.58731939694500820612134034625, 4.23751512314524123948264328204, 4.95635936344124367603610832251, 6.24817789524827216156353271254, 6.62593968191993972931457919863, 7.80977523614209372230442356427, 8.399824602320880588120919805353, 8.879425601906594055442109461641

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