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Properties

Label 1-151-151.25-r0-0-0
Degree $1$
Conductor $151$
Sign $0.868 + 0.495i$
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 + (0.968 − 0.248i)3-s + (0.913 + 0.406i)4-s + (−0.0209 + 0.999i)5-s + (−0.999 + 0.0418i)6-s + (0.944 + 0.328i)7-s + (−0.809 − 0.587i)8-s + (0.876 − 0.481i)9-s + (0.228 − 0.973i)10-s + (−0.699 + 0.714i)11-s + (0.985 + 0.166i)12-s + (0.146 + 0.989i)13-s + (−0.855 − 0.518i)14-s + (0.228 + 0.973i)15-s + (0.669 + 0.743i)16-s + (−0.756 − 0.653i)17-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.968 − 0.248i)3-s + (0.913 + 0.406i)4-s + (−0.0209 + 0.999i)5-s + (−0.999 + 0.0418i)6-s + (0.944 + 0.328i)7-s + (−0.809 − 0.587i)8-s + (0.876 − 0.481i)9-s + (0.228 − 0.973i)10-s + (−0.699 + 0.714i)11-s + (0.985 + 0.166i)12-s + (0.146 + 0.989i)13-s + (−0.855 − 0.518i)14-s + (0.228 + 0.973i)15-s + (0.669 + 0.743i)16-s + (−0.756 − 0.653i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.868 + 0.495i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.868 + 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003043365 + 0.2658256143i\)
\(L(\frac12)\) \(\approx\) \(1.003043365 + 0.2658256143i\)
\(L(1)\) \(\approx\) \(0.9811800198 + 0.1051497467i\)
\(L(1)\) \(\approx\) \(0.9811800198 + 0.1051497467i\)

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 \( 1 + (0.968 - 0.248i)T \)
5 \( 1 + (-0.0209 + 0.999i)T \)
7 \( 1 + (0.944 + 0.328i)T \)
11 \( 1 + (-0.699 + 0.714i)T \)
13 \( 1 + (0.146 + 0.989i)T \)
17 \( 1 + (-0.756 - 0.653i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (0.0627 + 0.998i)T \)
31 \( 1 + (0.387 - 0.921i)T \)
37 \( 1 + (0.783 - 0.621i)T \)
41 \( 1 + (0.535 + 0.844i)T \)
43 \( 1 + (0.944 - 0.328i)T \)
47 \( 1 + (0.463 - 0.886i)T \)
53 \( 1 + (-0.637 - 0.770i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.268 - 0.963i)T \)
67 \( 1 + (0.876 + 0.481i)T \)
71 \( 1 + (-0.756 + 0.653i)T \)
73 \( 1 + (-0.187 - 0.982i)T \)
79 \( 1 + (-0.992 - 0.125i)T \)
83 \( 1 + (-0.425 - 0.904i)T \)
89 \( 1 + (-0.570 + 0.821i)T \)
97 \( 1 + (-0.855 + 0.518i)T \)
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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

−27.64537349068520708897826815276, −27.111712126733679770807602515435, −26.08908025929326550822413194285, −25.16994807488012223365860145239, −24.2848416804028186085524402881, −23.721961904582027544965946039191, −21.42604949162212029076097707145, −20.87768901860609690474028228216, −19.95999600741830608500981916346, −19.26195727823595846459648464199, −17.87678452622669210774319363416, −17.11435574937722501831812736993, −15.77411415944193347378333823547, −15.28651542620943586615816901467, −13.87225478302949075237300634136, −12.82902959564576338772149051211, −11.17396214191338700529189328234, −10.27881413882369062803815707978, −8.9619222375749944444675503165, −8.27608283121443259391253898041, −7.605846624432167529067373109094, −5.716059364476231898024012331691, −4.363083514998298430703079531155, −2.59502935723659625949396043769, −1.22630800091025691940797545929, 1.97448900673199872402079443188, 2.55302501350382725941407845876, 4.21099650086640946026311472419, 6.501035107456973303701315335851, 7.4338120852720427856754997258, 8.32699942583052895937897748708, 9.392488885056784652805946260538, 10.51540876038481689608766755707, 11.498554891990209843006200694592, 12.74983179419266257486678279784, 14.293220354591845682056190883045, 14.977303687606682661863290928588, 16.000368338217662193640631479733, 17.61743961053759741189071495778, 18.44241788438390608452542543873, 18.893643185934268518947909483959, 20.18159039185270080082940190242, 20.94640889069152574697867991540, 21.823222944503121133033919033224, 23.52273232883220267403495576888, 24.591658301339579725087205815110, 25.47474929961155494236252050409, 26.33644823236693944258571783189, 26.885185340204957964851105109686, 27.97109182666283838973962422126

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