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Properties

Label 1-151-151.42-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.485 + 0.874i$
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  + (−0.104 − 0.994i)2-s + (−0.187 − 0.982i)3-s + (−0.978 + 0.207i)4-s + (0.146 − 0.989i)5-s + (−0.957 + 0.289i)6-s + (−0.699 + 0.714i)7-s + (0.309 + 0.951i)8-s + (−0.929 + 0.368i)9-s + (−0.999 − 0.0418i)10-s + (−0.756 − 0.653i)11-s + (0.387 + 0.921i)12-s + (−0.855 − 0.518i)13-s + (0.783 + 0.621i)14-s + (−0.999 + 0.0418i)15-s + (0.913 − 0.406i)16-s + (−0.268 + 0.963i)17-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.187 − 0.982i)3-s + (−0.978 + 0.207i)4-s + (0.146 − 0.989i)5-s + (−0.957 + 0.289i)6-s + (−0.699 + 0.714i)7-s + (0.309 + 0.951i)8-s + (−0.929 + 0.368i)9-s + (−0.999 − 0.0418i)10-s + (−0.756 − 0.653i)11-s + (0.387 + 0.921i)12-s + (−0.855 − 0.518i)13-s + (0.783 + 0.621i)14-s + (−0.999 + 0.0418i)15-s + (0.913 − 0.406i)16-s + (−0.268 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.485 + 0.874i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.485 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2184310190 - 0.3709152306i\)
\(L(\frac12)\) \(\approx\) \(-0.2184310190 - 0.3709152306i\)
\(L(1)\) \(\approx\) \(0.3278402740 - 0.5301644378i\)
\(L(1)\) \(\approx\) \(0.3278402740 - 0.5301644378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (-0.187 - 0.982i)T \)
5 \( 1 + (0.146 - 0.989i)T \)
7 \( 1 + (-0.699 + 0.714i)T \)
11 \( 1 + (-0.756 - 0.653i)T \)
13 \( 1 + (-0.855 - 0.518i)T \)
17 \( 1 + (-0.268 + 0.963i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.669 + 0.743i)T \)
29 \( 1 + (-0.425 - 0.904i)T \)
31 \( 1 + (-0.348 - 0.937i)T \)
37 \( 1 + (-0.0209 + 0.999i)T \)
41 \( 1 + (0.728 + 0.684i)T \)
43 \( 1 + (-0.699 - 0.714i)T \)
47 \( 1 + (0.228 - 0.973i)T \)
53 \( 1 + (-0.992 + 0.125i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.944 - 0.328i)T \)
67 \( 1 + (-0.929 - 0.368i)T \)
71 \( 1 + (-0.268 - 0.963i)T \)
73 \( 1 + (0.968 + 0.248i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.0627 - 0.998i)T \)
89 \( 1 + (-0.895 + 0.444i)T \)
97 \( 1 + (0.783 - 0.621i)T \)
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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

−28.633457070896520697561423794086, −27.15922245914093769760581240143, −26.72140994363353897568855803289, −25.994489346221707858586179620394, −25.13139184797220027984626513330, −23.564224589791583019704107851262, −22.7479535731531510409537766270, −22.31237483464008998023810637305, −21.03822980395725649685989763423, −19.71949679922893908002109847699, −18.49474641364724895036774809961, −17.54352171488047076257912601999, −16.48450791603018264698121441881, −15.80548367910761290278864257214, −14.631878579867717835098730540901, −14.10362725907505509002358697803, −12.61132304372409829767932645159, −10.823582128708787631133475094429, −10.006074758413966389339200895420, −9.254756179135579870792052957766, −7.4954161438952126998004320772, −6.73551247753168683767915455661, −5.40317116220394872590151260917, −4.28288371293591921554177047857, −2.99956052014800567445778061308, 0.376282232717851485749560592920, 1.99029790358285386786175731159, 3.09016993163124896679221146001, 5.0130721972837192346213921701, 5.90110258206780848458880093, 7.78522445720674686876674475640, 8.72042404722939872479186940450, 9.739726921323168624911631490894, 11.22027248777731434578594133698, 12.18786365779026608546610210180, 13.09330281841582755782864333953, 13.403361972618336595507638426110, 15.249299270804079389272497999933, 16.81964569322780004468402580523, 17.55892296122111429064800223893, 18.703922812836758838531816742656, 19.44249254030219857821207366002, 20.23538739746700679701658196402, 21.518975601882665730061647713151, 22.29091226911261098311902943026, 23.53854117003412631965970010079, 24.32825979545533496422525225206, 25.405103112532647122678675287850, 26.49413809067053510059958846119, 27.949542470925343589670904924614

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