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Properties

Label 1-155-155.123-r0-0-0
Degree $1$
Conductor $155$
Sign $-0.850 + 0.525i$
Analytic cond. $0.719816$
Root an. cond. $0.719816$
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  − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s − 11-s + i·12-s − i·13-s − 14-s + 16-s + i·17-s + i·18-s − 19-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s − 11-s + i·12-s − i·13-s − 14-s + 16-s + i·17-s + i·18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(0.719816\)
Root analytic conductor: \(0.719816\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 155,\ (0:\ ),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1905696120 - 0.6708330521i\)
\(L(\frac12)\) \(\approx\) \(-0.1905696120 - 0.6708330521i\)
\(L(1)\) \(\approx\) \(0.4051201693 - 0.6802082828i\)
\(L(1)\) \(\approx\) \(0.4051201693 - 0.6802082828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + iT \)
29 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + iT \)
43 \( 1 - iT \)
47 \( 1 - T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + iT \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 - iT \)
97 \( 1 + T \)
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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

−28.17211662977322629806166500864, −27.43723619824796783662528337845, −26.4184371226146375593393680550, −25.7213956486059833581118123091, −24.81897473349444485113321595307, −23.56761942662974807024007001748, −22.79914649169446955678761152403, −21.532977726036865971641965105231, −21.245996556899189546135638951676, −19.46013419908791834986580270743, −18.39352318233581070248771253944, −17.40362442770601317349060831187, −16.1480610008938394239380927770, −15.723528573301343544611843361448, −14.71861486531108909315247198465, −13.76916293382207518701506933802, −12.36976709003575976315570414201, −11.007486731808634269493723550613, −9.61003609657952765280663660973, −8.95516725340718248190334614511, −7.812683063676204263207574993662, −6.253732113049680601102045002286, −5.25218261120271345572723977553, −4.33573180831603522037596237238, −2.74404371985057509622896802737, 0.586970504022132361911899578933, 2.08784883756727384889564226685, 3.30990319380585093113374750272, 4.80640038930479320141127245221, 6.25483232105777009449609958373, 7.79823765659132201645538455617, 8.49944177387542857933306969792, 10.340405537058762176641962749819, 10.80281584597489183171604822352, 12.32723233269226879166822314566, 12.99830136245877962409424089468, 13.777408136160354048671452596317, 14.96036308663229962482598911721, 16.87937976177005941167722405363, 17.67258051367888520442306222818, 18.57757540601438216593974525538, 19.56447408884537929694533417447, 20.28138050411652026192214469599, 21.25242087540700703266462043759, 22.6638205079497722001610158893, 23.36395279320425142729331360843, 24.1053682531006644114141700200, 25.628382812958702276908663804123, 26.43235246352168322482523355209, 27.59520732858749554705082679345

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