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Properties

Label 1-151-151.62-r0-0-0
Degree $1$
Conductor $151$
Sign $-0.133 - 0.991i$
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 + (0.876 − 0.481i)3-s + (−0.978 + 0.207i)4-s + (0.463 + 0.886i)5-s + (−0.570 − 0.821i)6-s + (0.146 − 0.989i)7-s + (0.309 + 0.951i)8-s + (0.535 − 0.844i)9-s + (0.832 − 0.553i)10-s + (−0.855 + 0.518i)11-s + (−0.756 + 0.653i)12-s + (0.228 − 0.973i)13-s + (−0.999 − 0.0418i)14-s + (0.832 + 0.553i)15-s + (0.913 − 0.406i)16-s + (0.783 − 0.621i)17-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.876 − 0.481i)3-s + (−0.978 + 0.207i)4-s + (0.463 + 0.886i)5-s + (−0.570 − 0.821i)6-s + (0.146 − 0.989i)7-s + (0.309 + 0.951i)8-s + (0.535 − 0.844i)9-s + (0.832 − 0.553i)10-s + (−0.855 + 0.518i)11-s + (−0.756 + 0.653i)12-s + (0.228 − 0.973i)13-s + (−0.999 − 0.0418i)14-s + (0.832 + 0.553i)15-s + (0.913 − 0.406i)16-s + (0.783 − 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ -0.133 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8828748212 - 1.010020827i\)
\(L(\frac12)\) \(\approx\) \(0.8828748212 - 1.010020827i\)
\(L(1)\) \(\approx\) \(1.032037350 - 0.7126089693i\)
\(L(1)\) \(\approx\) \(1.032037350 - 0.7126089693i\)

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 \( 1 + (0.876 - 0.481i)T \)
5 \( 1 + (0.463 + 0.886i)T \)
7 \( 1 + (0.146 - 0.989i)T \)
11 \( 1 + (-0.855 + 0.518i)T \)
13 \( 1 + (0.228 - 0.973i)T \)
17 \( 1 + (0.783 - 0.621i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.669 + 0.743i)T \)
29 \( 1 + (-0.992 + 0.125i)T \)
31 \( 1 + (-0.268 + 0.963i)T \)
37 \( 1 + (-0.957 + 0.289i)T \)
41 \( 1 + (-0.425 + 0.904i)T \)
43 \( 1 + (0.146 + 0.989i)T \)
47 \( 1 + (0.996 - 0.0836i)T \)
53 \( 1 + (-0.187 + 0.982i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.0209 - 0.999i)T \)
67 \( 1 + (0.535 + 0.844i)T \)
71 \( 1 + (0.783 + 0.621i)T \)
73 \( 1 + (-0.929 + 0.368i)T \)
79 \( 1 + (0.968 + 0.248i)T \)
83 \( 1 + (-0.637 + 0.770i)T \)
89 \( 1 + (0.985 + 0.166i)T \)
97 \( 1 + (-0.999 + 0.0418i)T \)
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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

−28.08607660669815924198252204879, −27.20189853706833158558881846772, −26.07017748170248201238725945690, −25.49382452023557062180993074036, −24.506121011666971805317399752053, −23.95332957644620417145110282356, −22.35991479257059214069821980153, −21.29235394754312320081378275307, −20.77787388703308280431363569816, −18.98805281098359257645558447622, −18.58121242644916581915800879108, −16.927292394370558048154443142656, −16.24490409750862382879580025853, −15.3100942902818122424009471724, −14.32400002786025709382156915059, −13.41997513353971466496877083858, −12.39471673738054215238767272876, −10.362557159869992905376830029380, −9.20494071514278485751910644373, −8.6444088351742334785669512637, −7.70058441644359178613516378604, −5.87541721036712088938097345965, −5.09510747821619695942578729863, −3.761714606062268323516788223923, −1.91352802383519426684703898526, 1.32234062105609515399788580496, 2.758648146613883122746528323954, 3.45659880967375543722663505333, 5.15819099183400770362745779648, 7.14820275106197890949862332416, 7.86937247149976863245246331823, 9.40653596342159533465555785815, 10.24279587469823014901988342523, 11.16764152055854635795400063506, 12.74046997748130804077009182806, 13.4955973838594331692685572116, 14.23907189361135808266507287192, 15.36362426658619865724341762055, 17.3750984023327802778495874029, 18.09311275527496238034377577888, 18.892153020527122129032176479264, 20.03445047128069384655541521828, 20.629801686277400832383384741039, 21.60093608310059170367984856723, 22.9380353584421803416655435476, 23.53464759535061534054864934024, 25.18471540898731528756432315839, 26.10035460678582824318403816948, 26.674467769655772202368141874625, 27.75557451256091708789352407454

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