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Properties

Label 2-912-228.59-c0-0-0
Degree $2$
Conductor $912$
Sign $0.363 - 0.931i$
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  + (0.766 + 0.642i)3-s + (−1.11 + 0.642i)7-s + (0.173 + 0.984i)9-s + (0.439 + 0.524i)13-s + (0.5 + 0.866i)19-s + (−1.26 − 0.223i)21-s + (0.766 − 0.642i)25-s + (−0.500 + 0.866i)27-s + (0.173 + 0.300i)31-s − 1.96i·37-s + 0.684i·39-s + (−0.673 − 1.85i)43-s + (0.326 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (−1.76 − 0.642i)61-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (−1.11 + 0.642i)7-s + (0.173 + 0.984i)9-s + (0.439 + 0.524i)13-s + (0.5 + 0.866i)19-s + (−1.26 − 0.223i)21-s + (0.766 − 0.642i)25-s + (−0.500 + 0.866i)27-s + (0.173 + 0.300i)31-s − 1.96i·37-s + 0.684i·39-s + (−0.673 − 1.85i)43-s + (0.326 − 0.565i)49-s + (−0.173 + 0.984i)57-s + (−1.76 − 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.363 - 0.931i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155739663\)
\(L(\frac12)\) \(\approx\) \(1.155739663\)
\(L(1)\) 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 \( 1 + (-0.766 - 0.642i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.96iT - T^{2} \)
41 \( 1 + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \)
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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

−10.29751140044838181003372515374, −9.475359181056564897519919774634, −8.971901981826470917778346482464, −8.154762875238087233787601271124, −7.09117975733071958519193396551, −6.12019280026866542868229085820, −5.17166781731405271636363857033, −3.94567716652067600437599604543, −3.21362123084486307079913040451, −2.10856510065065962670930584359, 1.15741630604833331333751156162, 2.89034679860065617475391385705, 3.40817368674870713843355693494, 4.71136406018848628283843838057, 6.15299544050435080418855681501, 6.77643935112272604733710289559, 7.57428244983919508836667546422, 8.420918095431247227933722634904, 9.339487834319715711737943106764, 9.914205713540462480893395952009

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