Properties

Label 2-151-151.8-c1-0-4
Degree $2$
Conductor $151$
Sign $0.733 + 0.679i$
Analytic cond. $1.20574$
Root an. cond. $1.09806$
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  − 1.35·2-s + (−0.106 − 0.328i)3-s − 0.169·4-s + (−2.12 + 1.54i)5-s + (0.144 + 0.444i)6-s + (3.63 − 2.63i)7-s + 2.93·8-s + (2.33 − 1.69i)9-s + (2.86 − 2.08i)10-s + (0.100 − 0.309i)11-s + (0.0180 + 0.0555i)12-s + (−1.52 − 4.68i)13-s + (−4.91 + 3.57i)14-s + (0.732 + 0.531i)15-s − 3.63·16-s + (1.52 + 1.10i)17-s + ⋯
L(s)  = 1  − 0.956·2-s + (−0.0615 − 0.189i)3-s − 0.0845·4-s + (−0.948 + 0.689i)5-s + (0.0589 + 0.181i)6-s + (1.37 − 0.997i)7-s + 1.03·8-s + (0.776 − 0.564i)9-s + (0.907 − 0.659i)10-s + (0.0303 − 0.0932i)11-s + (0.00520 + 0.0160i)12-s + (−0.422 − 1.29i)13-s + (−1.31 + 0.954i)14-s + (0.189 + 0.137i)15-s − 0.908·16-s + (0.369 + 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(1.20574\)
Root analytic conductor: \(1.09806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :1/2),\ 0.733 + 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589363 - 0.231102i\)
\(L(\frac12)\) \(\approx\) \(0.589363 - 0.231102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 + (11.6 + 3.98i)T \)
good2 \( 1 + 1.35T + 2T^{2} \)
3 \( 1 + (0.106 + 0.328i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.12 - 1.54i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-3.63 + 2.63i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.100 + 0.309i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.52 + 4.68i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.52 - 1.10i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 - 1.61T + 19T^{2} \)
23 \( 1 - 4.76T + 23T^{2} \)
29 \( 1 + (0.175 - 0.539i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.471 - 0.342i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.95 - 6.00i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.72 + 11.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (4.65 + 3.38i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.08 - 3.33i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.48 - 7.63i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + (-2.47 + 7.62i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-11.0 - 8.03i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.1 - 7.36i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.40 + 2.47i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (12.8 - 9.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.77 + 4.91i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-5.81 - 4.22i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-12.2 - 8.86i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.82466712178962401870781893246, −11.55410776251748071298290708953, −10.68112847775857615214277892576, −10.03553966765213312283258601008, −8.505807235663374083295455463843, −7.58402562280259053831936732809, −7.19837102124035477082079961557, −4.93342539125232687099158126357, −3.70837758172452600595595511888, −1.04666673119604271068304856602, 1.65646899101635471162800539076, 4.47759386977805257556498603471, 4.97210831324541567764726740408, 7.31191305069890962852879040151, 8.114385327146470184812045694199, 8.895752921291847775381575949630, 9.809535790135718935277381337735, 11.22093250543346538237208874114, 11.78650558252377010277584503513, 12.96241526060106983901276507047

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