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Properties

Label 1-151-151.145-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: $\chi_{151} (145, \cdot )$
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

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

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