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Properties

Label 1-151-151.50-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.994 + 0.106i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (0.0627 − 0.998i)3-s + (−0.809 + 0.587i)4-s + (−0.992 − 0.125i)5-s + (0.968 − 0.248i)6-s + (−0.425 + 0.904i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (−0.187 − 0.982i)10-s + (0.0627 + 0.998i)11-s + (0.535 + 0.844i)12-s + (−0.637 + 0.770i)13-s + (−0.992 − 0.125i)14-s + (−0.187 + 0.982i)15-s + (0.309 − 0.951i)16-s + (−0.425 − 0.904i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.0627 − 0.998i)3-s + (−0.809 + 0.587i)4-s + (−0.992 − 0.125i)5-s + (0.968 − 0.248i)6-s + (−0.425 + 0.904i)7-s + (−0.809 − 0.587i)8-s + (−0.992 − 0.125i)9-s + (−0.187 − 0.982i)10-s + (0.0627 + 0.998i)11-s + (0.535 + 0.844i)12-s + (−0.637 + 0.770i)13-s + (−0.992 − 0.125i)14-s + (−0.187 + 0.982i)15-s + (0.309 − 0.951i)16-s + (−0.425 − 0.904i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.994 + 0.106i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.994 + 0.106i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02059863878 + 0.3849272675i\)
\(L(\frac12)\) \(\approx\) \(0.02059863878 + 0.3849272675i\)
\(L(1)\) \(\approx\) \(0.5960406471 + 0.3106834804i\)
\(L(1)\) \(\approx\) \(0.5960406471 + 0.3106834804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (0.0627 - 0.998i)T \)
5 \( 1 + (-0.992 - 0.125i)T \)
7 \( 1 + (-0.425 + 0.904i)T \)
11 \( 1 + (0.0627 + 0.998i)T \)
13 \( 1 + (-0.637 + 0.770i)T \)
17 \( 1 + (-0.425 - 0.904i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.929 + 0.368i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (-0.637 + 0.770i)T \)
41 \( 1 + (0.968 - 0.248i)T \)
43 \( 1 + (-0.425 - 0.904i)T \)
47 \( 1 + (0.968 - 0.248i)T \)
53 \( 1 + (0.535 - 0.844i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.0627 + 0.998i)T \)
67 \( 1 + (-0.992 + 0.125i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (-0.425 + 0.904i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (0.876 + 0.481i)T \)
89 \( 1 + (0.876 + 0.481i)T \)
97 \( 1 + (-0.992 + 0.125i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.699047259820000095532840096490, −26.63054262894778126949823391870, −26.39709360751436245189984397145, −24.29229321119312022431331755037, −23.30414733849769468933533578513, −22.50206743970068323652612154268, −21.72321507922370287793561000309, −20.59974856573382010059286087464, −19.66467944098298518883833317796, −19.31725708080734416623225026230, −17.56856491939188800302794040707, −16.46146952595014189798631155031, −15.32407949716882523318212013695, −14.451762381095618777615539758864, −13.27316006679343904456431284012, −12.078227849931389456560285790037, −10.756517668570557137074580458243, −10.56840129190740810507286127564, −9.08548728090355093797046022987, −8.01638231755624292536118229510, −6.08429885965407812811905733272, −4.548067503961937463778075060183, −3.81924598323705031488781034681, −2.84321889520642601034288252739, −0.284512986192346577420890993289, 2.37337012230610066275684742822, 3.98504650923568307019593427064, 5.32921539908300848403308564664, 6.663776796209741908680983045102, 7.391722898696479372813700999359, 8.45010704514678343201908626395, 9.44673447483205338541109787477, 11.847074566823582845460811575592, 12.22701864862648519729779737655, 13.30170031578252712534197669690, 14.580847319619088249785467165089, 15.344991714448161501264099111794, 16.42734292184635211018813115230, 17.51613391659521962804096345565, 18.605688986233043885058428027301, 19.25471243793547839040009547172, 20.54385926478743679456874726120, 22.190020342225771170033372166193, 22.86167183626468777537116712548, 23.81566942232917127420937947236, 24.52777082715858128024324224814, 25.42779385658113449950400398560, 26.22253046189899083524494600575, 27.53087431617007218789211031912, 28.37463951713907045649456031244

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