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Properties

Label 2-912-76.3-c1-0-11
Degree $2$
Conductor $912$
Sign $-0.756 + 0.654i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−0.939 + 0.342i)3-s + (−0.233 + 1.32i)5-s + (−3.20 + 1.85i)7-s + (0.766 − 0.642i)9-s + (1.31 + 0.761i)11-s + (0.286 − 0.788i)13-s + (−0.233 − 1.32i)15-s + (0.124 + 0.104i)17-s + (−4.35 + 0.0632i)19-s + (2.37 − 2.83i)21-s + (−1.85 + 0.327i)23-s + (2.99 + 1.08i)25-s + (−0.500 + 0.866i)27-s + (−6.75 − 8.04i)29-s + (−2.95 − 5.11i)31-s + ⋯
L(s)  = 1  + (−0.542 + 0.197i)3-s + (−0.104 + 0.593i)5-s + (−1.21 + 0.699i)7-s + (0.255 − 0.214i)9-s + (0.397 + 0.229i)11-s + (0.0795 − 0.218i)13-s + (−0.0604 − 0.342i)15-s + (0.0301 + 0.0253i)17-s + (−0.999 + 0.0144i)19-s + (0.519 − 0.618i)21-s + (−0.387 + 0.0683i)23-s + (0.598 + 0.217i)25-s + (−0.0962 + 0.166i)27-s + (−1.25 − 1.49i)29-s + (−0.529 − 0.917i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.756 + 0.654i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.756 + 0.654i)\)

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 \( 1 \)
3 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (4.35 - 0.0632i)T \)
good5 \( 1 + (0.233 - 1.32i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (3.20 - 1.85i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.31 - 0.761i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.286 + 0.788i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.124 - 0.104i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.85 - 0.327i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (6.75 + 8.04i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.95 + 5.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.67iT - 37T^{2} \)
41 \( 1 + (-0.788 - 2.16i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-2.89 - 0.509i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (5.63 + 6.71i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (6.69 - 1.18i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (9.06 + 7.60i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.741 - 4.20i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.57 - 7.19i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.29 - 7.34i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-6.53 + 2.37i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-12.9 + 4.72i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.134 - 0.0775i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.92 + 5.30i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.19 - 6.19i)T + (-16.8 - 95.5i)T^{2} \)
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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

−9.669354250083560337119568046306, −9.266737679157787718803356172529, −8.045231959069330782980767763557, −6.99331311262689752702060031786, −6.23472781486245066676329783818, −5.67542842354804605396262379701, −4.28394311923491398094827925748, −3.36996761943869037902109411058, −2.20060200646501097585989898107, 0, 1.45900034544631415661747605981, 3.22267913488333027096086592229, 4.17384759474136590357962098935, 5.16953868280838298440198678018, 6.32481132800055020138652584909, 6.78207470245733646618676331246, 7.80642186994927938569612620020, 8.920241319906959327426879531906, 9.493077500038322582648215232607

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