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Properties

Label 1-115-115.43-r0-0-0
Degree 11
Conductor 115115
Sign 0.630+0.775i0.630 + 0.775i
Analytic cond. 0.5340570.534057
Root an. cond. 0.5340570.534057
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(115s/2 ΓR(s) L(s)=((0.630+0.775i) Λ‾(1−s)\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}
Λ(s)=(115s/2 ΓR(s) L(s)=((0.630+0.775i) Λ‾(1−s)\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: 11
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.630+0.775i0.630 + 0.775i
Analytic conductor: 0.5340570.534057
Root analytic conductor: 0.5340570.534057
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ115(43,⋅)\chi_{115} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 115, (0: ), 0.630+0.775i)(1,\ 115,\ (0:\ ),\ 0.630 + 0.775i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.316206088+0.6261912862i1.316206088 + 0.6261912862i
L(12)L(\frac12) ≈\approx 1.316206088+0.6261912862i1.316206088 + 0.6261912862i
L(1)L(1) ≈\approx 1.327026719+0.4711818198i1.327026719 + 0.4711818198i
L(1)L(1) ≈\approx 1.327026719+0.4711818198i1.327026719 + 0.4711818198i

Euler product

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