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Properties

Label 1-115-115.43-r0-0-0
Degree $1$
Conductor $115$
Sign $0.630 + 0.775i$
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  + (0.281 + 0.959i)2-s + (0.755 − 0.654i)3-s + (−0.841 + 0.540i)4-s + (0.841 + 0.540i)6-s + (0.909 + 0.415i)7-s + (−0.755 − 0.654i)8-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.281 + 0.959i)12-s + (−0.909 + 0.415i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.540 + 0.841i)17-s + (0.989 − 0.142i)18-s + (0.841 − 0.540i)19-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (0.755 − 0.654i)3-s + (−0.841 + 0.540i)4-s + (0.841 + 0.540i)6-s + (0.909 + 0.415i)7-s + (−0.755 − 0.654i)8-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (−0.281 + 0.959i)12-s + (−0.909 + 0.415i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.540 + 0.841i)17-s + (0.989 − 0.142i)18-s + (0.841 − 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.630 + 0.775i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.630 + 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.316206088 + 0.6261912862i\)
\(L(\frac12)\) \(\approx\) \(1.316206088 + 0.6261912862i\)
\(L(1)\) \(\approx\) \(1.327026719 + 0.4711818198i\)
\(L(1)\) \(\approx\) \(1.327026719 + 0.4711818198i\)

Euler product

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

−29.38743942231747391959292111236, −27.90444523527028730776837860645, −27.18120845126125234967976853407, −26.6364175937870286315174692198, −24.94418814700682804867205542642, −24.10421926152272284099882389839, −22.476908796682333485990085499, −21.953406168986866542463394961818, −20.63005114125577277486420181158, −20.23514890062341506454452359030, −19.17435597676157785396057390882, −17.91698598673362896325592906551, −16.6687438978735729803460902686, −15.01090206078581225572747598382, −14.31175152059120910341696206345, −13.452863014209357144519789441787, −11.902533977467241110139654060056, −10.93581750766475183702089268742, −9.79810530168422645291673436315, −8.89076805141742171507742001248, −7.56309293766689314391110748523, −5.28645035421431937151903606951, −4.294834849531520828766601204149, −3.13293286203268126209951938529, −1.68882732753112917358808210043, 1.90255129529334251237256984808, 3.6970464799716643271301241928, 5.05336663635275158752696516091, 6.58666546180564797311136482428, 7.49916724058087610368157288267, 8.614728206503530756767327948338, 9.44892524229902589203473808122, 11.71630298155003136317519883663, 12.66185241262694910648187026097, 13.94905254861002214279566181579, 14.63634626524071947618338816190, 15.44781686231376687456988808949, 17.119574799345009321355486269880, 17.8112399032890438423827690948, 18.93990305668895033419324384857, 20.07718955946683115338301679158, 21.425793258437842153619415614, 22.33986940914778452002433139381, 23.81083469412424826572378502711, 24.442235664726670596484940896, 25.07802696981514283180462154230, 26.25060990636746178252500640917, 27.014896952824111901520552768771, 28.22506526651277211722216046814, 29.84471482873281633682317421321

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