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Properties

Label 2-152-152.75-c5-0-13
Degree $2$
Conductor $152$
Sign $-0.407 + 0.913i$
Analytic cond. $24.3783$
Root an. cond. $4.93744$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.05 − 2.54i)2-s + 27.1i·3-s + (19.0 + 25.7i)4-s + 36.9i·5-s + (69.3 − 137. i)6-s + 150. i·7-s + (−30.4 − 178. i)8-s − 496.·9-s + (94.2 − 186. i)10-s − 603.·11-s + (−700. + 517. i)12-s + 566.·13-s + (383. − 760. i)14-s − 1.00e3·15-s + (−301. + 978. i)16-s − 1.98e3·17-s + ⋯
L(s)  = 1  + (−0.892 − 0.450i)2-s + 1.74i·3-s + (0.594 + 0.804i)4-s + 0.661i·5-s + (0.785 − 1.55i)6-s + 1.16i·7-s + (−0.168 − 0.985i)8-s − 2.04·9-s + (0.298 − 0.590i)10-s − 1.50·11-s + (−1.40 + 1.03i)12-s + 0.929·13-s + (0.523 − 1.03i)14-s − 1.15·15-s + (−0.294 + 0.955i)16-s − 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.407 + 0.913i$
Analytic conductor: \(24.3783\)
Root analytic conductor: \(4.93744\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :5/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6044116482\)
\(L(\frac12)\) \(\approx\) \(0.6044116482\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.05 + 2.54i)T \)
19 \( 1 + (-1.30e3 - 873. i)T \)
good3 \( 1 - 27.1iT - 243T^{2} \)
5 \( 1 - 36.9iT - 3.12e3T^{2} \)
7 \( 1 - 150. iT - 1.68e4T^{2} \)
11 \( 1 + 603.T + 1.61e5T^{2} \)
13 \( 1 - 566.T + 3.71e5T^{2} \)
17 \( 1 + 1.98e3T + 1.41e6T^{2} \)
23 \( 1 - 1.46e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.74e3T + 2.05e7T^{2} \)
31 \( 1 - 1.90e3T + 2.86e7T^{2} \)
37 \( 1 + 9.12e3T + 6.93e7T^{2} \)
41 \( 1 - 2.64e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.00e4T + 1.47e8T^{2} \)
47 \( 1 - 2.04e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.00e4T + 4.18e8T^{2} \)
59 \( 1 - 1.51e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.82e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.59e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.98e4T + 1.80e9T^{2} \)
73 \( 1 - 2.15e4T + 2.07e9T^{2} \)
79 \( 1 - 7.27e4T + 3.07e9T^{2} \)
83 \( 1 - 1.46e4T + 3.93e9T^{2} \)
89 \( 1 + 1.86e4iT - 5.58e9T^{2} \)
97 \( 1 - 2.09e4iT - 8.58e9T^{2} \)
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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

−12.37911298685500966684165128972, −11.00490015584782316113555891705, −10.87732746613439108851155493184, −9.748903855820241839426304045497, −8.943575656550541940732184079928, −8.078546205799334046176846244625, −6.26151922537574415745567493618, −4.93722418385127907035612434869, −3.37166238570712623160465799155, −2.51938984947819091136794924018, 0.30969363395015919155773475236, 1.07945710680372202359616797332, 2.46032295453414391201326014090, 5.07775944089694037409044986522, 6.49282016897404268422370075577, 7.17712041923479439104388048531, 8.140564498026870010111298421550, 8.792546325451440891865801572519, 10.48786714094306946991995422811, 11.24673796578092434935119046499

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