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Properties

Label 1-115-115.9-r0-0-0
Degree $1$
Conductor $115$
Sign $0.899 - 0.436i$
Analytic cond. $0.534057$
Root an. cond. $0.534057$
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  + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (0.415 + 0.909i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.841 − 0.540i)12-s + (0.654 − 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.415 − 0.909i)17-s + (0.959 − 0.281i)18-s + (0.415 − 0.909i)19-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (0.415 + 0.909i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.841 − 0.540i)12-s + (0.654 − 0.755i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.415 − 0.909i)17-s + (0.959 − 0.281i)18-s + (0.415 − 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.899 - 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7856604236 - 0.1804260319i\)
\(L(\frac12)\) \(\approx\) \(0.7856604236 - 0.1804260319i\)
\(L(1)\) \(\approx\) \(0.8180778615 - 0.08416917071i\)
\(L(1)\) \(\approx\) \(0.8180778615 - 0.08416917071i\)

Euler product

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

−29.09504377747457773612588702187, −28.1977602578746538498079289171, −27.17012667641668344649882407908, −26.715442317084237390408958014423, −25.71802475239812542413208176388, −24.52058918337506156539545323696, −23.02569368150788084970834923724, −21.70775536182886827856374998687, −21.09805415609550202257733841662, −20.09902906969487533137594473732, −19.26374637759563873552787371606, −17.8244870721594373379827712085, −16.82399057921902084673749438258, −16.17652899594132054037517288535, −14.68905437364405221080602580705, −13.59491797397308054911976702909, −11.77288736852062093315966234963, −11.001252298762170207139936722640, −10.05261744148212146220344224674, −8.886948899367759420174498172904, −8.02735713365346133728826599281, −6.35174382566979969212532565746, −4.32582690502585084405507868571, −3.48509220920111840723768480619, −1.56993199084183246467085806944, 1.23436887100098870283896134822, 2.57125067082364513623865501846, 5.15798189805529112523312172141, 6.37599754950528362553895776249, 7.41383535561467292782302159614, 8.50115723824680880940427037450, 9.34987942404269768373705515086, 11.12490865907018464928229476686, 11.96666094280271764733273680893, 13.50778380727481700386783823063, 14.65453270387981796173804824219, 15.54276169504265082896351181351, 17.03336579711744830958652562079, 18.03340958235673012880172907915, 18.43273603321969681700848748829, 19.806959280999838279076182854, 20.45027365014625314055167073091, 22.310421117374462087783444085988, 23.46894513548036163085415635160, 24.49179557944433519968937497129, 25.093931826658543044622214392788, 25.882120943806390944198842765046, 27.30662577385304499010332171944, 28.089120431594442560546437638275, 29.03781609835574455186863481054

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