Properties

Label 1-115-115.53-r0-0-0
Degree $1$
Conductor $115$
Sign $0.660 + 0.750i$
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  + (−0.989 + 0.142i)2-s + (0.909 + 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.959 − 0.281i)6-s + (0.540 + 0.841i)7-s + (−0.909 + 0.415i)8-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (0.989 + 0.142i)12-s + (−0.540 + 0.841i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (−0.755 − 0.654i)18-s + (−0.959 + 0.281i)19-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (0.909 + 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.959 − 0.281i)6-s + (0.540 + 0.841i)7-s + (−0.909 + 0.415i)8-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (0.989 + 0.142i)12-s + (−0.540 + 0.841i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (0.281 − 0.959i)17-s + (−0.755 − 0.654i)18-s + (−0.959 + 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8694626792 + 0.3932016493i\)
\(L(\frac12)\) \(\approx\) \(0.8694626792 + 0.3932016493i\)
\(L(1)\) \(\approx\) \(0.9164537822 + 0.2515906542i\)
\(L(1)\) \(\approx\) \(0.9164537822 + 0.2515906542i\)

Euler product

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

−29.24868316281304453222654483777, −27.88148217654275451384492021039, −27.09057228346832598446913067475, −26.07557398465779278391825074742, −25.36730515862420039222636273359, −24.36916357136370930028006292590, −23.36821764503148520751854098729, −21.51793202246640004964653551436, −20.449243932780904198109635448245, −19.8960228712334716644856702869, −18.92402067827591210706520083563, −17.66400663006251668435018277372, −17.10564717216873379902281835683, −15.34689728179198392477184050941, −14.68424711236582380699786912134, −13.12907104904845266012932469480, −12.119111673466837265675171035513, −10.54564311775213465142881928352, −9.735538400734983142373779787385, −8.31723914683589480375771447760, −7.64438336852594964825485760586, −6.54864059029530069314503404892, −4.212490045634931749103453321454, −2.63148708374185400467866722854, −1.35935495493342975762268341344, 1.87659792705625067946234597722, 3.05553626557108661557684707681, 4.98651830979259253733518907458, 6.62050869083954223890536049354, 8.071614427245782656743130766750, 8.780217125157839056552536672787, 9.72640584563349213455630906856, 11.0070996076235675803018574303, 12.12650894604414077973051030136, 14.00916712229031535426135148898, 14.84145181038762540991908393249, 15.92000011782317600845700503438, 16.78461656108867334112036124322, 18.29599559745132649562213511374, 19.03717081106329787092545877519, 19.92570049612389676999965630108, 21.23194685235200732402978679179, 21.63759958927262408308853019604, 23.74078842522074811726279281007, 24.866516448760993029923721143622, 25.28032162277294495395627601022, 26.68013877666362745575891447586, 27.10541221524557860854560944668, 28.13060915270144626283741619835, 29.27576167450832828113385397773

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