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Properties

Label 1-155-155.74-r1-0-0
Degree $1$
Conductor $155$
Sign $-0.789 + 0.613i$
Analytic cond. $16.6570$
Root an. cond. $16.6570$
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 + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 0.866i)6-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (−0.913 − 0.406i)11-s + (−0.978 − 0.207i)12-s + (−0.978 + 0.207i)13-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)16-s + (0.913 − 0.406i)17-s + (0.978 − 0.207i)18-s + (−0.978 − 0.207i)19-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 0.866i)6-s + (0.104 + 0.994i)7-s + (0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (−0.913 − 0.406i)11-s + (−0.978 − 0.207i)12-s + (−0.978 + 0.207i)13-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)16-s + (0.913 − 0.406i)17-s + (0.978 − 0.207i)18-s + (−0.978 − 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.789 + 0.613i$
Analytic conductor: \(16.6570\)
Root analytic conductor: \(16.6570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (1:\ ),\ -0.789 + 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1659915227 + 0.4839077609i\)
\(L(\frac12)\) \(\approx\) \(0.1659915227 + 0.4839077609i\)
\(L(1)\) \(\approx\) \(0.7844081233 + 0.03648834274i\)
\(L(1)\) \(\approx\) \(0.7844081233 + 0.03648834274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.978 - 0.207i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.104 + 0.994i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 - T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−26.98379171782266692255290550636, −26.10165966566807891615433973814, −25.53456327973514391653010616415, −24.39680201381284417369203168063, −23.619829204453434988702576082161, −23.0272145485296933766610227930, −21.36727543975682712059445503184, −20.05186311293813800809706840180, −19.33694217866574118754907968943, −18.21519135729782672330346605837, −17.42697996119895009299831925430, −16.42023206780244988846129702665, −15.02105219365573370645000362468, −14.3868026466026719211469132191, −13.32413996529220051790943127281, −12.483601719905840122373927910734, −10.48441437947295882183809018519, −9.59750009941365442618243786598, −8.10503814437735563217660624659, −7.598722631645725185016563974394, −6.58777378785796643181592402094, −5.14090229913295392912859798729, −3.67852245929681084767500150744, −1.78545217430197520597921150682, −0.18480987410351101718740814062, 2.20134696646311288612405937041, 2.92892596619617608981019531016, 4.391581302095237558351196678745, 5.44905087396813113549648273231, 7.79999394157138319220108446524, 8.61121235996971484713168396015, 9.65801874184507641155935619214, 10.45919830783088346779969582515, 11.68956737136416131458303151120, 12.727758803669778735516013817993, 13.93284903771484381416877573343, 14.93643028747311419384893153016, 16.09823076335394688372227426589, 17.21703935677832297821119529210, 18.64838156920064755094714750617, 19.14251301924040901315171028899, 20.34453682630448152359559192591, 21.23114611122742592862021056544, 21.76329480992708968063670266251, 22.768977666567238997494081636449, 24.31924462279507032117735148988, 25.57145517576921168490194295395, 26.29769890971167006165505795543, 27.287048134035929320164226350199, 27.99982081703532631418472315257

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