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Properties

Label 2-151-151.32-c1-0-7
Degree $2$
Conductor $151$
Sign $0.242 + 0.970i$
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  + (−0.793 − 1.37i)2-s + 2.75·3-s + (−0.259 + 0.449i)4-s + (−0.154 − 0.267i)5-s + (−2.18 − 3.79i)6-s + (0.365 + 0.632i)7-s − 2.35·8-s + 4.61·9-s + (−0.244 + 0.424i)10-s + (0.331 + 0.573i)11-s + (−0.715 + 1.24i)12-s + (−1.84 + 3.19i)13-s + (0.579 − 1.00i)14-s + (−0.425 − 0.737i)15-s + (2.38 + 4.12i)16-s + (−1.46 − 2.52i)17-s + ⋯
L(s)  = 1  + (−0.561 − 0.971i)2-s + 1.59·3-s + (−0.129 + 0.224i)4-s + (−0.0690 − 0.119i)5-s + (−0.893 − 1.54i)6-s + (0.138 + 0.239i)7-s − 0.831·8-s + 1.53·9-s + (−0.0774 + 0.134i)10-s + (0.0998 + 0.172i)11-s + (−0.206 + 0.357i)12-s + (−0.510 + 0.884i)13-s + (0.154 − 0.268i)14-s + (−0.109 − 0.190i)15-s + (0.596 + 1.03i)16-s + (−0.354 − 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(151\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(1.20574\)
Root analytic conductor: \(1.09806\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 151,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03636 - 0.809280i\)
\(L(\frac12)\) \(\approx\) \(1.03636 - 0.809280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 + (-10.7 - 5.97i)T \)
good2 \( 1 + (0.793 + 1.37i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 2.75T + 3T^{2} \)
5 \( 1 + (0.154 + 0.267i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.365 - 0.632i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.331 - 0.573i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.84 - 3.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.46 + 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.43T + 19T^{2} \)
23 \( 1 + (0.974 + 1.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.72T + 29T^{2} \)
31 \( 1 + (-3.89 - 6.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.17 - 8.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.69T + 41T^{2} \)
43 \( 1 + (-0.139 + 0.241i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.313 - 0.542i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.49T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + (-2.69 + 4.67i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 5.81T + 67T^{2} \)
71 \( 1 + (-6.51 + 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 - 6.42T + 79T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 + (2.70 - 4.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.51 + 9.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.66679044603065692103645772988, −11.77672339613991774899486135105, −10.57263574034285325269779744584, −9.503480266694845511409711681125, −8.969806335523315554640439935263, −8.030634686620255929489255289418, −6.60942317910314684705646926348, −4.43245302197048759207309254656, −2.92139567477966469377324654621, −1.94932297307457183596859752276, 2.58571279868572295236154079783, 3.92136650324484510140704611764, 5.95617989176920665970991149556, 7.44585194080894221044949266532, 7.84810099124407407694636515330, 8.883978594407161331950696982846, 9.585097741193557708247196984550, 10.96184555460885895146137324750, 12.59817629748159869954525364813, 13.39787968831569316902737188252

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