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Properties

Label 1-115-115.34-r1-0-0
Degree $1$
Conductor $115$
Sign $-0.874 + 0.484i$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
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  + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.654 + 0.755i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (−0.841 − 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (−0.841 + 0.540i)18-s + (0.654 + 0.755i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.654 + 0.755i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (−0.841 − 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (−0.841 + 0.540i)18-s + (0.654 + 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.874 + 0.484i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ -0.874 + 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3846464350 + 1.488823357i\)
\(L(\frac12)\) \(\approx\) \(0.3846464350 + 1.488823357i\)
\(L(1)\) \(\approx\) \(0.8173070897 + 0.7165280942i\)
\(L(1)\) \(\approx\) \(0.8173070897 + 0.7165280942i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.415 + 0.909i)T \)
3 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (-0.142 + 0.989i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (0.142 + 0.989i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (0.841 + 0.540i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.654 + 0.755i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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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

−28.746412399273006402101788587527, −27.511657386170970497695968821171, −26.55265610032366696734063602697, −25.91325636166233862396893752485, −24.784270451101712966329217998685, −23.34019885754658444648895397467, −22.35826477930638822870785933211, −20.90397539612136233056946216753, −20.18809479553393707820469220009, −19.69712258065015311045419948847, −18.25652158938001909501747084770, −17.65154014606970173487040364360, −16.08424904023660353553200685299, −14.6958126155036682099724240172, −13.30459491271099802936385941339, −12.99457763790864917174668127589, −11.37040759465903527595568756439, −10.10105605131777349691615199733, −9.32835925746378491995924826038, −7.89981766178031711825593464466, −7.18923564946508697562248662215, −4.65226891176544538708269060125, −3.42067833458685208587460394937, −2.241190881511097093464521271931, −0.66850315480766078520556507633, 1.85983193771982388485765438264, 3.63182096262474508837638189188, 5.18382125748187886212408669349, 6.455304719274252285264289468797, 7.915841642664692121940814045907, 8.79036600616910015335598447666, 9.561859019640675184609498881740, 10.9780722556769648874097330212, 12.87376056959988513355531068852, 13.985596844318259508144471541264, 14.863354211293740252405836893202, 15.85824866280751234561263690222, 16.58496160678897181577579379718, 18.34836190827409998948653415742, 18.85144646548292925771328865283, 19.928027603034534646618492272015, 21.40975218939759540740703302086, 22.16838552415276389505220170695, 23.82724155136288746929918621327, 24.54241868082960637599210571176, 25.507615568102246072335626437036, 26.31282826688820906735318026633, 27.08668326690310590682357391215, 28.19666344525965017036569817042, 29.19541195422272384963192706490

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