Properties

Label 1-115-115.59-r0-0-0
Degree $1$
Conductor $115$
Sign $-0.763 + 0.645i$
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  + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.959 + 0.281i)6-s + (−0.841 − 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.142 − 0.989i)11-s + (0.142 + 0.989i)12-s + (−0.841 + 0.540i)13-s + (−0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (0.654 + 0.755i)18-s + (−0.959 − 0.281i)19-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.959 + 0.281i)6-s + (−0.841 − 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.142 − 0.989i)11-s + (0.142 + 0.989i)12-s + (−0.841 + 0.540i)13-s + (−0.654 + 0.755i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (0.654 + 0.755i)18-s + (−0.959 − 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ -0.763 + 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1737697020 - 0.4744073259i\)
\(L(\frac12)\) \(\approx\) \(-0.1737697020 - 0.4744073259i\)
\(L(1)\) \(\approx\) \(0.3655983593 - 0.5648328054i\)
\(L(1)\) \(\approx\) \(0.3655983593 - 0.5648328054i\)

Euler product

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

−29.91251445117234843757131223693, −28.47749777361771353977882995816, −27.77951085883021788081186618819, −26.77569064798373578517849717849, −25.72617872601928320488017640698, −25.1369333475357650818278596128, −23.56456171506906329011964647672, −22.78812810717119679610621968064, −22.0441815803955969058140375624, −21.02535200837310346450670231830, −19.51233554727832059644640005801, −18.16328649987554478305568306103, −17.08712384856090488750259714382, −16.339300644782667354249413272707, −15.15312314541193174765563285138, −14.76447561566892746881576616893, −12.93557103605278499517401628236, −12.10615430259548954171339808560, −10.12883862142128522372320169790, −9.5531060010117707021057253274, −8.15461655604027509192788652825, −6.66183388275861022440370953755, −5.588883945366500606863540564003, −4.552929029078934027361505056759, −3.16859264678050938064899146447, 0.47558893852744344433186668546, 2.21333934851078255977198639131, 3.5750096551382545313452001800, 5.2492721155096988228333109384, 6.515247531156862128892888865517, 7.95004242825062538050822115583, 9.399592244960054538594066160, 10.63291199469799348982001650010, 11.647932489145365215042368362650, 12.696421904960526181566728546520, 13.4757697023180688555879249396, 14.4678002781719172115118497329, 16.52850205180166255560588936234, 17.306205854884359344971841795327, 18.86110938681255219161582020244, 19.08702050167291933966900396294, 20.23452078780113598254409321582, 21.59629626490291203239624812586, 22.50362174128530211120514363718, 23.46116027160754739830336312976, 24.20479719123811144858550093615, 25.71870072167066909853987683210, 26.81732468610646606362843612202, 27.98177064886252402939388126882, 29.03422639432898916788090851358

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