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Properties

Label 2-912-228.167-c0-0-0
Degree $2$
Conductor $912$
Sign $0.290 - 0.956i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)3-s + (1.70 + 0.984i)7-s + (−0.939 + 0.342i)9-s + (−1.26 − 0.223i)13-s + (0.5 − 0.866i)19-s + (−0.673 + 1.85i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 1.62i)31-s − 0.684i·37-s − 1.28i·39-s + (0.439 + 0.524i)43-s + (1.43 + 2.49i)49-s + (0.939 + 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (1.70 + 0.984i)7-s + (−0.939 + 0.342i)9-s + (−1.26 − 0.223i)13-s + (0.5 − 0.866i)19-s + (−0.673 + 1.85i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (−0.939 + 1.62i)31-s − 0.684i·37-s − 1.28i·39-s + (0.439 + 0.524i)43-s + (1.43 + 2.49i)49-s + (0.939 + 0.342i)57-s + (−1.17 − 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.290 - 0.956i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.290 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152624207\)
\(L(\frac12)\) \(\approx\) \(1.152624207\)
\(L(1)\) 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 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 0.684iT - T^{2} \)
41 \( 1 + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
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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

−10.57767207810050101909272485832, −9.500598180011310915991170971751, −8.867424539573938647751305445297, −8.135055110129959143783768891249, −7.30255710366752172288600260223, −5.79037378220534759411041259039, −4.95677267093251883962064193656, −4.60422785778914165055794953683, −3.02827478829573137930144099920, −2.07073964914053771211321183476, 1.30744447492846902158167675822, 2.28345782723894083283013187983, 3.80858564929013575462270485963, 4.89644599135766081528685512061, 5.74558259754925626221270877700, 7.13445117138140669533216337503, 7.53418948751378611104820923853, 8.134534622279837945419560485031, 9.175598661458003818553823050123, 10.19083814258200102374031024581

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