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Properties

Label 1-151-151.10-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: $\chi_{151} (10, \cdot )$
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

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

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