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Properties

Label 1-151-151.69-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.531 - 0.847i$
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 + (0.535 − 0.844i)3-s + (0.913 − 0.406i)4-s + (−0.570 + 0.821i)5-s + (−0.348 + 0.937i)6-s + (−0.957 − 0.289i)7-s + (−0.809 + 0.587i)8-s + (−0.425 − 0.904i)9-s + (0.387 − 0.921i)10-s + (0.463 − 0.886i)11-s + (0.146 − 0.989i)12-s + (−0.895 − 0.444i)13-s + (0.996 + 0.0836i)14-s + (0.387 + 0.921i)15-s + (0.669 − 0.743i)16-s + (0.228 − 0.973i)17-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.535 − 0.844i)3-s + (0.913 − 0.406i)4-s + (−0.570 + 0.821i)5-s + (−0.348 + 0.937i)6-s + (−0.957 − 0.289i)7-s + (−0.809 + 0.587i)8-s + (−0.425 − 0.904i)9-s + (0.387 − 0.921i)10-s + (0.463 − 0.886i)11-s + (0.146 − 0.989i)12-s + (−0.895 − 0.444i)13-s + (0.996 + 0.0836i)14-s + (0.387 + 0.921i)15-s + (0.669 − 0.743i)16-s + (0.228 − 0.973i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.531 - 0.847i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.531 - 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2284366628 - 0.4128143497i\)
\(L(\frac12)\) \(\approx\) \(0.2284366628 - 0.4128143497i\)
\(L(1)\) \(\approx\) \(0.5516954585 - 0.2100565294i\)
\(L(1)\) \(\approx\) \(0.5516954585 - 0.2100565294i\)

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 \( 1 + (0.535 - 0.844i)T \)
5 \( 1 + (-0.570 + 0.821i)T \)
7 \( 1 + (-0.957 - 0.289i)T \)
11 \( 1 + (0.463 - 0.886i)T \)
13 \( 1 + (-0.895 - 0.444i)T \)
17 \( 1 + (0.228 - 0.973i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (0.968 - 0.248i)T \)
31 \( 1 + (-0.855 - 0.518i)T \)
37 \( 1 + (0.832 - 0.553i)T \)
41 \( 1 + (-0.637 - 0.770i)T \)
43 \( 1 + (-0.957 + 0.289i)T \)
47 \( 1 + (0.985 - 0.166i)T \)
53 \( 1 + (-0.929 - 0.368i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.999 + 0.0418i)T \)
67 \( 1 + (-0.425 + 0.904i)T \)
71 \( 1 + (0.228 + 0.973i)T \)
73 \( 1 + (0.728 - 0.684i)T \)
79 \( 1 + (0.876 + 0.481i)T \)
83 \( 1 + (-0.187 - 0.982i)T \)
89 \( 1 + (0.944 + 0.328i)T \)
97 \( 1 + (0.996 - 0.0836i)T \)
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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

−28.3325242039299450740330458870, −27.36018271934824149695181494835, −26.66258016352028987784866023235, −25.48584882773113627820749384819, −25.07117223911418263152421205427, −23.614309100028923518743670996329, −22.15942617452661690914366988777, −21.284261235373957220720619943680, −20.07853795992634904497073055612, −19.73424574549130172761152469529, −18.77470267095991267820355877948, −16.9664901129481276794961588970, −16.61370953095911609417775813240, −15.50012670769659301883450361277, −14.71549299596366478906877261045, −12.71210673620290386040123693744, −12.077512008361738121723104670648, −10.5119684193107861277896340578, −9.6471803139553103063606938567, −8.846176814430675838933104449411, −7.90201547620774717514968763736, −6.47191338103741911821039526085, −4.61146555983553571866575800541, −3.452952641985731224882400680909, −1.99997264314392252183873103213, 0.49053416107113822339889283408, 2.50419011030176039748477844273, 3.38773523017351541335420856089, 6.04814377458962052072331510975, 6.99544614068716662963744646831, 7.65914524678744828316149678597, 8.91163708358687360079760260993, 9.95115462442790039061873490157, 11.24507672796740660691484427352, 12.20305857087974441715644264089, 13.65016621005835321377138496015, 14.72750948465630692412623269034, 15.680403757994383976122475614494, 16.86486480448154046710659389457, 17.96287465798511733509474076116, 18.9683248948252236029406150228, 19.486076922813356394473649135150, 20.14602781256930849474274336180, 21.837286304842237337726910017222, 23.15763126479518368725831268979, 23.97408990543632611364719177323, 25.13161185502167195524971154715, 25.7499589423164137407986335543, 26.773742769754367095753436634, 27.32243564817032324403957868256

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