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Properties

Label 2-152-152.59-c1-0-2
Degree $2$
Conductor $152$
Sign $-0.206 - 0.978i$
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  + (−1.39 − 0.245i)2-s + (−0.950 + 1.13i)3-s + (1.87 + 0.684i)4-s + (1.60 − 1.34i)6-s + (−2.44 − 1.41i)8-s + (0.141 + 0.801i)9-s + (−3.20 + 5.55i)11-s + (−2.56 + 1.47i)12-s + (3.06 + 2.57i)16-s + (−0.329 + 1.87i)17-s − 1.15i·18-s + (−0.511 + 4.32i)19-s + (5.83 − 6.94i)22-s + (3.92 − 1.43i)24-s + (3.83 − 3.21i)25-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.548 + 0.653i)3-s + (0.939 + 0.342i)4-s + (0.653 − 0.548i)6-s + (−0.866 − 0.500i)8-s + (0.0471 + 0.267i)9-s + (−0.966 + 1.67i)11-s + (−0.739 + 0.426i)12-s + (0.766 + 0.642i)16-s + (−0.0799 + 0.453i)17-s − 0.271i·18-s + (−0.117 + 0.993i)19-s + (1.24 − 1.48i)22-s + (0.802 − 0.291i)24-s + (0.766 − 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.206 - 0.978i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324311 + 0.400029i\)
\(L(\frac12)\) \(\approx\) \(0.324311 + 0.400029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.245i)T \)
19 \( 1 + (0.511 - 4.32i)T \)
good3 \( 1 + (0.950 - 1.13i)T + (-0.520 - 2.95i)T^{2} \)
5 \( 1 + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.20 - 5.55i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.329 - 1.87i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (-5.51 + 6.57i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-11.6 + 4.22i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-6.87 - 1.21i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (3.13 - 0.553i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-7.18 - 6.02i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (7.97 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.20 - 9.77i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (19.0 + 3.36i)T + (91.1 + 33.1i)T^{2} \)
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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

−12.81927481898259212156593013870, −12.15595918455977806739945831509, −10.78560151342982971424869390528, −10.35332303585707946577923360856, −9.468713081723191240228173578730, −8.090474989290259908430111993269, −7.18548469676738484248260379279, −5.69745075433866692396778644121, −4.30491674583522624818876642621, −2.24230261540818247704408082937, 0.72557688006535973967890410672, 2.89922478770229076294881519981, 5.45315305847081634603459376762, 6.41385718638754390643953773474, 7.43765568256135194961758639674, 8.515704866787628689204537884650, 9.502941596935127890124843908438, 10.97275897724031763497757736935, 11.29342053219698115137888569511, 12.58456228637961357853475332652

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