Properties

Label 2-912-76.51-c1-0-8
Degree $2$
Conductor $912$
Sign $0.944 - 0.327i$
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  + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11698 + 0.356765i\)
\(L(\frac12)\) \(\approx\) \(2.11698 + 0.356765i\)
\(L(\frac{3}{2})\) 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

−9.983680287689499624931312212745, −9.144516544889139529899777924570, −8.489026825822248793019899499994, −7.84970760359577721088621718879, −6.91238773105931880189142470622, −5.34002982421341085511330747644, −5.04083877349462980400541246632, −3.84827666655221956893589492222, −2.55877759232529329467915660013, −1.42452957051676808160622173652, 1.18665425438903847191621187816, 2.55519829918482498309272213770, 3.59609981640647122666358399276, 4.65059682982139770125851951021, 5.65285825341140638875953896837, 6.91230765352779981113242844653, 7.62704026023687478814582723183, 8.134519810095323496481442998309, 9.168337791262111515838128531136, 10.11761105454689276631361952214

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