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Properties

Label 2-152-152.125-c1-0-0
Degree $2$
Conductor $152$
Sign $-0.459 - 0.888i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
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  + (−1.20 + 0.747i)2-s + (−2.77 − 1.60i)3-s + (0.883 − 1.79i)4-s + (0.667 + 0.385i)5-s + (4.53 − 0.149i)6-s − 2.17·7-s + (0.279 + 2.81i)8-s + (3.64 + 6.32i)9-s + (−1.08 + 0.0360i)10-s + 2.54i·11-s + (−5.33 + 3.56i)12-s + (−3.71 + 2.14i)13-s + (2.60 − 1.62i)14-s + (−1.23 − 2.14i)15-s + (−2.43 − 3.17i)16-s + (−0.358 + 0.620i)17-s + ⋯
L(s)  = 1  + (−0.849 + 0.528i)2-s + (−1.60 − 0.926i)3-s + (0.441 − 0.897i)4-s + (0.298 + 0.172i)5-s + (1.85 − 0.0612i)6-s − 0.821·7-s + (0.0989 + 0.995i)8-s + (1.21 + 2.10i)9-s + (−0.344 + 0.0113i)10-s + 0.767i·11-s + (−1.53 + 1.03i)12-s + (−1.02 + 0.594i)13-s + (0.697 − 0.433i)14-s + (−0.319 − 0.553i)15-s + (−0.609 − 0.792i)16-s + (−0.0868 + 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.459 - 0.888i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.459 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118385 + 0.194525i\)
\(L(\frac12)\) \(\approx\) \(0.118385 + 0.194525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.747i)T \)
19 \( 1 + (-2.77 - 3.36i)T \)
good3 \( 1 + (2.77 + 1.60i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.667 - 0.385i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.17T + 7T^{2} \)
11 \( 1 - 2.54iT - 11T^{2} \)
13 \( 1 + (3.71 - 2.14i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.358 - 0.620i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.87 - 6.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.06 - 0.612i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.83T + 31T^{2} \)
37 \( 1 - 2.90iT - 37T^{2} \)
41 \( 1 + (-0.940 + 1.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.25 + 0.725i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.53 + 2.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.00 - 2.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.25 + 1.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.16 - 2.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.4 + 6.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.90 - 13.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.82 - 6.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.20 + 5.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.0802iT - 83T^{2} \)
89 \( 1 + (-4.36 - 7.55i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.83 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.07382064988640787512923792283, −12.14448913777213556206400912496, −11.32181804564560785037557029849, −10.19011636855509704978749343663, −9.497256165312429454663213884439, −7.54700190369428370331758703060, −6.99832396200897494674870066623, −6.03660337001060909514625262112, −5.10378464842956343554849981224, −1.78458560666887456507366818557, 0.34369764309337049582967745119, 3.28907804230224870121407212937, 4.90611346848309314140072846109, 6.11710935229583798858282134281, 7.23202218857317819060816989320, 9.140056492702559465570936148882, 9.727729995216147528638984960078, 10.70472389927898901924790713852, 11.30883141111373917852479115755, 12.39210106530347854189362549862

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