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Properties

Label 1-151-151.136-r0-0-0
Degree $1$
Conductor $151$
Sign $0.999 + 0.0185i$
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  + (0.913 + 0.406i)2-s + (−0.187 − 0.982i)3-s + (0.669 + 0.743i)4-s + (0.783 + 0.621i)5-s + (0.228 − 0.973i)6-s + (−0.268 − 0.963i)7-s + (0.309 + 0.951i)8-s + (−0.929 + 0.368i)9-s + (0.463 + 0.886i)10-s + (0.944 − 0.328i)11-s + (0.604 − 0.796i)12-s + (−0.0209 + 0.999i)13-s + (0.146 − 0.989i)14-s + (0.463 − 0.886i)15-s + (−0.104 + 0.994i)16-s + (−0.699 − 0.714i)17-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.187 − 0.982i)3-s + (0.669 + 0.743i)4-s + (0.783 + 0.621i)5-s + (0.228 − 0.973i)6-s + (−0.268 − 0.963i)7-s + (0.309 + 0.951i)8-s + (−0.929 + 0.368i)9-s + (0.463 + 0.886i)10-s + (0.944 − 0.328i)11-s + (0.604 − 0.796i)12-s + (−0.0209 + 0.999i)13-s + (0.146 − 0.989i)14-s + (0.463 − 0.886i)15-s + (−0.104 + 0.994i)16-s + (−0.699 − 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.999 + 0.0185i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.999 + 0.0185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.895867970 + 0.01762178127i\)
\(L(\frac12)\) \(\approx\) \(1.895867970 + 0.01762178127i\)
\(L(1)\) \(\approx\) \(1.720014600 + 0.03476286711i\)
\(L(1)\) \(\approx\) \(1.720014600 + 0.03476286711i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
3 \( 1 + (-0.187 - 0.982i)T \)
5 \( 1 + (0.783 + 0.621i)T \)
7 \( 1 + (-0.268 - 0.963i)T \)
11 \( 1 + (0.944 - 0.328i)T \)
13 \( 1 + (-0.0209 + 0.999i)T \)
17 \( 1 + (-0.699 - 0.714i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.978 + 0.207i)T \)
29 \( 1 + (-0.425 - 0.904i)T \)
31 \( 1 + (0.985 + 0.166i)T \)
37 \( 1 + (-0.855 - 0.518i)T \)
41 \( 1 + (0.728 + 0.684i)T \)
43 \( 1 + (-0.268 + 0.963i)T \)
47 \( 1 + (-0.957 + 0.289i)T \)
53 \( 1 + (-0.992 + 0.125i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.756 - 0.653i)T \)
67 \( 1 + (-0.929 - 0.368i)T \)
71 \( 1 + (-0.699 + 0.714i)T \)
73 \( 1 + (0.968 + 0.248i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.0627 - 0.998i)T \)
89 \( 1 + (0.832 + 0.553i)T \)
97 \( 1 + (0.146 + 0.989i)T \)
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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

−28.15430322100755600761818172013, −27.583053069171753856596056170794, −25.81736440367884787311948300609, −25.04540784857877609450907180774, −24.17677594201619527535452935688, −22.48418544834634948743898243037, −22.31664756005691817503319556846, −21.274924728926395555589155971185, −20.46088460324284189572963157503, −19.62134259704397622803051775600, −17.9877173437497137832525436392, −16.813620326581894619601879359322, −15.76026009553353235722855531832, −14.935617854395583013284782722461, −13.93729337198352344074619698009, −12.58541784347491341501986939255, −11.9368503995859519090295616345, −10.49204739796000340974819378686, −9.70200521528115349266434730343, −8.63294009692272288632120915708, −6.242193289145983694799870707344, −5.61795379516010834733210030990, −4.52653715886918753615433327049, −3.28786824749074029489860048177, −1.845478252253069589344441093733, 1.77673467394124759795275003186, 3.084669830515981714627461095561, 4.58580619458758816403356517753, 6.25254169883231590013716718592, 6.643595276142551770575873017774, 7.64573027362396822349482212605, 9.3268404267776309687216122876, 11.060975514839733335652755647885, 11.76181837587419962665460657523, 13.2244310341270483690312702902, 13.83666028226475991999096959111, 14.38487154534945686082178194744, 16.07899016251911334649752970960, 17.15091964013283851381956029464, 17.76133743813052713093672453897, 19.24050914786754973449755154022, 20.1369047289193580312836302020, 21.493203019211230676978175805459, 22.446192222319015865323396372591, 23.08376335167007405586790825553, 24.28688708979280221031279295393, 24.75759645477692816578380144675, 26.0539574988492129864919564872, 26.4835351449562177925596601558, 28.529558891375550934385838088631

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