Properties

Label 2-912-76.75-c1-0-2
Degree $2$
Conductor $912$
Sign $0.114 - 0.993i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  − 3-s − 3.37·5-s − 0.792i·7-s + 9-s − 0.792i·11-s − 5.04i·13-s + 3.37·15-s + 5.37·17-s + (−4 + 1.73i)19-s + 0.792i·21-s + 8.51i·23-s + 6.37·25-s − 27-s + 10.0i·29-s − 8.74·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.50·5-s − 0.299i·7-s + 0.333·9-s − 0.238i·11-s − 1.40i·13-s + 0.870·15-s + 1.30·17-s + (−0.917 + 0.397i)19-s + 0.172i·21-s + 1.77i·23-s + 1.27·25-s − 0.192·27-s + 1.87i·29-s − 1.57·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.114 - 0.993i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.114 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424336 + 0.378157i\)
\(L(\frac12)\) \(\approx\) \(0.424336 + 0.378157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + 3.37T + 5T^{2} \)
7 \( 1 + 0.792iT - 7T^{2} \)
11 \( 1 + 0.792iT - 11T^{2} \)
13 \( 1 + 5.04iT - 13T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
23 \( 1 - 8.51iT - 23T^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 + 8.74T + 31T^{2} \)
37 \( 1 + 5.04iT - 37T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 - 9.30iT - 43T^{2} \)
47 \( 1 - 4.25iT - 47T^{2} \)
53 \( 1 + 3.16iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 5.37T + 61T^{2} \)
67 \( 1 - 9.48T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 4.11T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + 13.2iT - 89T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−10.55485100624293765874571116242, −9.548343033430614883443187772432, −8.341903421944930656544637237173, −7.69244088283222367184622452531, −7.14839097121117454128937685430, −5.79677153032710855568665066029, −5.10485937446315768105789629432, −3.81026836997235737790763411114, −3.30120663720140239022297223840, −1.11186385128181932382564624403, 0.35143735854871599649762726020, 2.24759979608745054011809640632, 3.87052911292979782953202490879, 4.33359954585422204844628530313, 5.47645914520473065222480776230, 6.63740818499600890420239376467, 7.25546198935187917360656634540, 8.224312407863223103464847712898, 8.914589279638326545119700472105, 10.03433111311526134307336233548

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