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Properties

Label 2-966-161.19-c1-0-22
Degree $2$
Conductor $966$
Sign $0.982 + 0.186i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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  + (0.995 + 0.0950i)2-s + (−0.690 + 0.723i)3-s + (0.981 + 0.189i)4-s + (0.631 − 0.325i)5-s + (−0.755 + 0.654i)6-s + (2.64 − 0.152i)7-s + (0.959 + 0.281i)8-s + (−0.0475 − 0.998i)9-s + (0.659 − 0.263i)10-s + (−0.617 − 6.46i)11-s + (−0.814 + 0.580i)12-s + (1.45 + 0.209i)13-s + (2.64 + 0.0997i)14-s + (−0.200 + 0.681i)15-s + (0.928 + 0.371i)16-s + (0.342 − 0.989i)17-s + ⋯
L(s)  = 1  + (0.703 + 0.0672i)2-s + (−0.398 + 0.417i)3-s + (0.490 + 0.0946i)4-s + (0.282 − 0.145i)5-s + (−0.308 + 0.267i)6-s + (0.998 − 0.0574i)7-s + (0.339 + 0.0996i)8-s + (−0.0158 − 0.332i)9-s + (0.208 − 0.0834i)10-s + (−0.186 − 1.94i)11-s + (−0.235 + 0.167i)12-s + (0.403 + 0.0580i)13-s + (0.706 + 0.0266i)14-s + (−0.0516 + 0.175i)15-s + (0.232 + 0.0929i)16-s + (0.0830 − 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.982 + 0.186i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.982 + 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50604 - 0.235500i\)
\(L(\frac12)\) \(\approx\) \(2.50604 - 0.235500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 - 0.0950i)T \)
3 \( 1 + (0.690 - 0.723i)T \)
7 \( 1 + (-2.64 + 0.152i)T \)
23 \( 1 + (-1.29 + 4.61i)T \)
good5 \( 1 + (-0.631 + 0.325i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (0.617 + 6.46i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (-1.45 - 0.209i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-0.342 + 0.989i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (-0.0500 - 0.144i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-1.12 - 1.29i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-1.65 + 0.401i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (4.22 - 0.201i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-0.907 + 1.41i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-1.59 - 5.41i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (-10.3 - 5.97i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.73 + 6.02i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (-4.72 - 11.7i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (5.82 - 5.55i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (6.00 + 4.27i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (-4.45 - 9.75i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-0.725 + 3.76i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-0.501 + 0.638i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (2.23 - 1.43i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-1.27 + 5.24i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (5.80 + 3.73i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.32771895723907401249993847846, −8.993388912743018954782145177615, −8.407673847949748817504023578543, −7.40722574063712306082174085224, −6.16922789473641520989296851841, −5.64518688066885096430585192362, −4.80514445599228861574666452618, −3.83190241403991008496029475665, −2.76711958900995807485243404179, −1.12280330042653576634031823504, 1.58650087418330298722349984006, 2.35418255671303577496824275732, 3.98710734440916179737739305465, 4.86202525631279773607863187719, 5.56481677946286314067526750494, 6.60805231569267143589329288676, 7.39433063989035503782843225363, 8.065161319111456235880562124181, 9.343794793892600922768347429040, 10.30673761532476679050757238207

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