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Properties

Label 1-151-151.138-r0-0-0
Degree $1$
Conductor $151$
Sign $0.371 - 0.928i$
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  + (0.669 + 0.743i)2-s + (−0.637 − 0.770i)3-s + (−0.104 + 0.994i)4-s + (−0.756 − 0.653i)5-s + (0.146 − 0.989i)6-s + (0.387 − 0.921i)7-s + (−0.809 + 0.587i)8-s + (−0.187 + 0.982i)9-s + (−0.0209 − 0.999i)10-s + (−0.348 − 0.937i)11-s + (0.832 − 0.553i)12-s + (−0.268 − 0.963i)13-s + (0.944 − 0.328i)14-s + (−0.0209 + 0.999i)15-s + (−0.978 − 0.207i)16-s + (0.604 − 0.796i)17-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (−0.637 − 0.770i)3-s + (−0.104 + 0.994i)4-s + (−0.756 − 0.653i)5-s + (0.146 − 0.989i)6-s + (0.387 − 0.921i)7-s + (−0.809 + 0.587i)8-s + (−0.187 + 0.982i)9-s + (−0.0209 − 0.999i)10-s + (−0.348 − 0.937i)11-s + (0.832 − 0.553i)12-s + (−0.268 − 0.963i)13-s + (0.944 − 0.328i)14-s + (−0.0209 + 0.999i)15-s + (−0.978 − 0.207i)16-s + (0.604 − 0.796i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.371 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7507806512 - 0.5083699242i\)
\(L(\frac12)\) \(\approx\) \(0.7507806512 - 0.5083699242i\)
\(L(1)\) \(\approx\) \(0.9600322000 - 0.1404428618i\)
\(L(1)\) \(\approx\) \(0.9600322000 - 0.1404428618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.669 + 0.743i)T \)
3 \( 1 + (-0.637 - 0.770i)T \)
5 \( 1 + (-0.756 - 0.653i)T \)
7 \( 1 + (0.387 - 0.921i)T \)
11 \( 1 + (-0.348 - 0.937i)T \)
13 \( 1 + (-0.268 - 0.963i)T \)
17 \( 1 + (0.604 - 0.796i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (-0.570 - 0.821i)T \)
37 \( 1 + (-0.699 + 0.714i)T \)
41 \( 1 + (-0.929 + 0.368i)T \)
43 \( 1 + (0.387 + 0.921i)T \)
47 \( 1 + (0.783 + 0.621i)T \)
53 \( 1 + (0.0627 - 0.998i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.985 + 0.166i)T \)
67 \( 1 + (-0.187 - 0.982i)T \)
71 \( 1 + (0.604 + 0.796i)T \)
73 \( 1 + (-0.992 + 0.125i)T \)
79 \( 1 + (-0.425 - 0.904i)T \)
83 \( 1 + (0.728 + 0.684i)T \)
89 \( 1 + (0.228 - 0.973i)T \)
97 \( 1 + (0.944 + 0.328i)T \)
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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

−28.24695244436999054572688055836, −27.589301746363431361214047606841, −26.669449704464705130398331475025, −25.292764980223087040986792780186, −23.63693010649132316784217862208, −23.32120276005057964063551676620, −22.20683650393888670534279497472, −21.47185034795626527608261246600, −20.71702302601232522555838111882, −19.30381006101269404451328704766, −18.5841146159627977002824559729, −17.33760259430366291023063285286, −15.75961585700692901190804430501, −15.06106367569773777607446853284, −14.40122972953511359202893117338, −12.42280875085358547917958542126, −11.953332108239367088154758939307, −10.90124967449462443665422784318, −10.08591752696239715241809266009, −8.80060460138397968977485544934, −6.89692362115835227118037582992, −5.63277340329661375709152147087, −4.56203198435842221185363580835, −3.58268502032679049574167658101, −2.097100615665542960572673276619, 0.68671485541656639358922981159, 3.103606060667939463014118324847, 4.65370926051976312162981428862, 5.425341685075634687673586886428, 6.87619326286419291961570642425, 7.73928959960788482329922675857, 8.52641884805441075949816769937, 10.7978019578664967559853358242, 11.718230885258338162064771144264, 12.8454135817607812365053889329, 13.4336954755932458715432582926, 14.693382171206581110968580477110, 16.02946208042190998338957215798, 16.7607462087817635804864748323, 17.528374348810225356049858001951, 18.78881277263378270747368484263, 20.056947576047448677161980083003, 21.065843972027020338901757240094, 22.42821912930726856396949721412, 23.27308384114053809654311456517, 23.90790049545896756188539736048, 24.54021351433592493398442240463, 25.59875242324014602499078423081, 27.06838059501056428505095681403, 27.56826900395791454736572294495

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