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Properties

Label 1-151-151.110-r0-0-0
Degree $1$
Conductor $151$
Sign $0.817 - 0.576i$
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  + (0.309 − 0.951i)2-s + (0.968 + 0.248i)3-s + (−0.809 − 0.587i)4-s + (0.876 + 0.481i)5-s + (0.535 − 0.844i)6-s + (−0.187 + 0.982i)7-s + (−0.809 + 0.587i)8-s + (0.876 + 0.481i)9-s + (0.728 − 0.684i)10-s + (0.968 − 0.248i)11-s + (−0.637 − 0.770i)12-s + (−0.929 + 0.368i)13-s + (0.876 + 0.481i)14-s + (0.728 + 0.684i)15-s + (0.309 + 0.951i)16-s + (−0.187 − 0.982i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.968 + 0.248i)3-s + (−0.809 − 0.587i)4-s + (0.876 + 0.481i)5-s + (0.535 − 0.844i)6-s + (−0.187 + 0.982i)7-s + (−0.809 + 0.587i)8-s + (0.876 + 0.481i)9-s + (0.728 − 0.684i)10-s + (0.968 − 0.248i)11-s + (−0.637 − 0.770i)12-s + (−0.929 + 0.368i)13-s + (0.876 + 0.481i)14-s + (0.728 + 0.684i)15-s + (0.309 + 0.951i)16-s + (−0.187 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.817 - 0.576i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.817 - 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.666266831 - 0.5286946992i\)
\(L(\frac12)\) \(\approx\) \(1.666266831 - 0.5286946992i\)
\(L(1)\) \(\approx\) \(1.529804459 - 0.4332483816i\)
\(L(1)\) \(\approx\) \(1.529804459 - 0.4332483816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.968 + 0.248i)T \)
5 \( 1 + (0.876 + 0.481i)T \)
7 \( 1 + (-0.187 + 0.982i)T \)
11 \( 1 + (0.968 - 0.248i)T \)
13 \( 1 + (-0.929 + 0.368i)T \)
17 \( 1 + (-0.187 - 0.982i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.0627 - 0.998i)T \)
31 \( 1 + (-0.992 - 0.125i)T \)
37 \( 1 + (-0.929 + 0.368i)T \)
41 \( 1 + (0.535 - 0.844i)T \)
43 \( 1 + (-0.187 - 0.982i)T \)
47 \( 1 + (0.535 - 0.844i)T \)
53 \( 1 + (-0.637 + 0.770i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.968 - 0.248i)T \)
67 \( 1 + (0.876 - 0.481i)T \)
71 \( 1 + (-0.187 + 0.982i)T \)
73 \( 1 + (-0.187 + 0.982i)T \)
79 \( 1 + (-0.992 + 0.125i)T \)
83 \( 1 + (-0.425 + 0.904i)T \)
89 \( 1 + (-0.425 + 0.904i)T \)
97 \( 1 + (0.876 - 0.481i)T \)
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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

−27.794278093859680167964253962400, −26.861504522202993001492975146973, −25.84739426115873281427775339543, −25.31539546321688802203295036096, −24.37084752808489281181951122684, −23.65363477549112209155352152431, −22.24029615001559574209030228897, −21.394001791739024588643280169538, −20.21701540271412724371287912970, −19.35878996173706787088825854967, −17.787299877821191552590319894192, −17.16489489625037977445936062109, −16.13889212484319743936114329221, −14.559693608084145357910283606828, −14.33434656827900036066613038171, −13.04719352249646191529890762324, −12.55374046751548079824093208306, −10.131687962584497346180109479173, −9.295664711704204399072717161769, −8.215676186607258539609423809099, −7.13001512399020458228798928853, −6.14507463756044134424456131856, −4.55650456753726521027022696981, −3.54779734740597207718136432617, −1.66997018205881040886966919890, 2.07372194511266887793853205304, 2.61086455348422083545196969256, 4.01985210133293824777476779310, 5.3708239312924479984175590001, 6.777622583738587225389858893977, 8.767110368019619195601900728341, 9.36736465872816995354799133974, 10.25106817377766177315310384434, 11.63231947837802417816830310572, 12.73242917443577792816951761884, 13.90000618151823010243733489960, 14.46103340719220562039442034596, 15.45375723947879107346840963039, 17.20082356779196609551812404909, 18.49038621995869300458797587539, 19.11353354416022232813321464945, 20.09999407453674727436471949917, 21.171874972743718535016014418241, 21.98574878352263152685582683341, 22.3630822749994276316087711980, 24.2753398600915664397005281297, 25.02583194494991578023388270800, 26.10925692970110001256128617594, 27.07382451389539644662227633091, 28.00294954014865782978603104318

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