Properties

Label 1-151-151.22-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.689 - 0.723i$
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  + (−0.978 − 0.207i)2-s + (−0.929 − 0.368i)3-s + (0.913 + 0.406i)4-s + (−0.957 + 0.289i)5-s + (0.832 + 0.553i)6-s + (−0.0209 + 0.999i)7-s + (−0.809 − 0.587i)8-s + (0.728 + 0.684i)9-s + (0.996 − 0.0836i)10-s + (0.146 − 0.989i)11-s + (−0.699 − 0.714i)12-s + (0.463 − 0.886i)13-s + (0.228 − 0.973i)14-s + (0.996 + 0.0836i)15-s + (0.669 + 0.743i)16-s + (−0.855 + 0.518i)17-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (−0.929 − 0.368i)3-s + (0.913 + 0.406i)4-s + (−0.957 + 0.289i)5-s + (0.832 + 0.553i)6-s + (−0.0209 + 0.999i)7-s + (−0.809 − 0.587i)8-s + (0.728 + 0.684i)9-s + (0.996 − 0.0836i)10-s + (0.146 − 0.989i)11-s + (−0.699 − 0.714i)12-s + (0.463 − 0.886i)13-s + (0.228 − 0.973i)14-s + (0.996 + 0.0836i)15-s + (0.669 + 0.743i)16-s + (−0.855 + 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.689 - 0.723i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.689 - 0.723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07862216794 - 0.1835085744i\)
\(L(\frac12)\) \(\approx\) \(0.07862216794 - 0.1835085744i\)
\(L(1)\) \(\approx\) \(0.3658949759 - 0.08651050323i\)
\(L(1)\) \(\approx\) \(0.3658949759 - 0.08651050323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T \)
3 \( 1 + (-0.929 - 0.368i)T \)
5 \( 1 + (-0.957 + 0.289i)T \)
7 \( 1 + (-0.0209 + 0.999i)T \)
11 \( 1 + (0.146 - 0.989i)T \)
13 \( 1 + (0.463 - 0.886i)T \)
17 \( 1 + (-0.855 + 0.518i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.637 - 0.770i)T \)
31 \( 1 + (-0.756 - 0.653i)T \)
37 \( 1 + (-0.999 + 0.0418i)T \)
41 \( 1 + (0.0627 - 0.998i)T \)
43 \( 1 + (-0.0209 - 0.999i)T \)
47 \( 1 + (-0.895 + 0.444i)T \)
53 \( 1 + (0.968 + 0.248i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.783 + 0.621i)T \)
67 \( 1 + (0.728 - 0.684i)T \)
71 \( 1 + (-0.855 - 0.518i)T \)
73 \( 1 + (0.876 - 0.481i)T \)
79 \( 1 + (-0.187 - 0.982i)T \)
83 \( 1 + (-0.992 + 0.125i)T \)
89 \( 1 + (0.604 + 0.796i)T \)
97 \( 1 + (0.228 + 0.973i)T \)
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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

−28.09258096308663510337754462004, −27.598959853406580352475425597116, −26.64951495680198012536711409268, −25.88527895579675071752822825005, −24.3004641555061756484066817393, −23.57748161610285212199946380906, −22.92666590711455547498302550527, −21.34324670112580262266412512383, −20.2439054403392994894007969145, −19.564638747301483192104661396706, −18.22404998956019886137811545927, −17.35772113626819444241830203130, −16.47126905213038473272431714585, −15.78868538206125278875226495189, −14.76293600518741860538318774439, −12.879620657980386832067177641379, −11.54251364969828915729855597193, −11.05706215590756794276981935326, −9.85738622065330567135712144261, −8.79315001112937188127642113211, −7.20970811761770648866085467706, −6.77952257275050207844951935047, −4.94954085723133052620314287137, −3.86538756534997324514450057542, −1.43492804052548860762972029704, 0.27289106241351366616593162200, 2.162919509640195230211599429404, 3.74573391228325807729146849270, 5.7500203560456984412022967928, 6.62771759192211768944200829286, 7.999317469332936247605912351576, 8.7129124364364236357933145314, 10.51016066415003575049898024188, 11.14065699393113947172877452225, 12.0675499462725210423298586838, 12.913930695804166446048218131783, 15.078990793550767698553022733460, 15.85842339179758474137064545353, 16.76931926919329196451859036749, 17.9040241748405767154908520885, 18.82218042521145192498101656189, 19.20997067980085201352105467907, 20.64435845991322841490931048308, 21.89941922660407065759872590535, 22.66950190304273223288166331486, 24.08237538859994067938218221921, 24.63153555131122451137283621106, 25.89635046564905449310939777186, 27.098675323057245370453188417143, 27.75349920421192377702987838806

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