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Properties

Label 2-115-115.107-c1-0-7
Degree 22
Conductor 115115
Sign 0.364+0.931i0.364 + 0.931i
Analytic cond. 0.9182790.918279
Root an. cond. 0.9582690.958269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 − 1.01i)2-s + (−1.01 − 0.377i)3-s + (0.237 − 0.810i)4-s + (0.440 − 2.19i)5-s + (−1.74 + 0.513i)6-s + (2.09 − 0.456i)7-s + (0.680 + 1.82i)8-s + (−1.38 − 1.20i)9-s + (−1.62 − 3.40i)10-s + (−0.856 + 0.123i)11-s + (−0.546 + 0.730i)12-s + (−1.18 + 5.43i)13-s + (2.37 − 2.73i)14-s + (−1.27 + 2.05i)15-s + (4.18 + 2.69i)16-s + (−2.77 + 5.08i)17-s + ⋯
L(s)  = 1  + (0.954 − 0.714i)2-s + (−0.584 − 0.217i)3-s + (0.118 − 0.405i)4-s + (0.197 − 0.980i)5-s + (−0.713 + 0.209i)6-s + (0.792 − 0.172i)7-s + (0.240 + 0.645i)8-s + (−0.461 − 0.400i)9-s + (−0.512 − 1.07i)10-s + (−0.258 + 0.0371i)11-s + (−0.157 + 0.210i)12-s + (−0.327 + 1.50i)13-s + (0.633 − 0.731i)14-s + (−0.328 + 0.529i)15-s + (1.04 + 0.672i)16-s + (−0.673 + 1.23i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.364+0.931i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+1/2) L(s)=((0.364+0.931i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.364+0.931i0.364 + 0.931i
Analytic conductor: 0.9182790.918279
Root analytic conductor: 0.9582690.958269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ115(107,⋅)\chi_{115} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :1/2), 0.364+0.931i)(2,\ 115,\ (\ :1/2),\ 0.364 + 0.931i)

Particular Values

L(1)L(1) ≈\approx 1.17788−0.803548i1.17788 - 0.803548i
L(12)L(\frac12) ≈\approx 1.17788−0.803548i1.17788 - 0.803548i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−0.440+2.19i)T 1 + (-0.440 + 2.19i)T
23 1+(−0.530+4.76i)T 1 + (-0.530 + 4.76i)T
good2 1+(−1.35+1.01i)T+(0.563−1.91i)T2 1 + (-1.35 + 1.01i)T + (0.563 - 1.91i)T^{2}
3 1+(1.01+0.377i)T+(2.26+1.96i)T2 1 + (1.01 + 0.377i)T + (2.26 + 1.96i)T^{2}
7 1+(−2.09+0.456i)T+(6.36−2.90i)T2 1 + (-2.09 + 0.456i)T + (6.36 - 2.90i)T^{2}
11 1+(0.856−0.123i)T+(10.5−3.09i)T2 1 + (0.856 - 0.123i)T + (10.5 - 3.09i)T^{2}
13 1+(1.18−5.43i)T+(−11.8−5.40i)T2 1 + (1.18 - 5.43i)T + (-11.8 - 5.40i)T^{2}
17 1+(2.77−5.08i)T+(−9.19−14.3i)T2 1 + (2.77 - 5.08i)T + (-9.19 - 14.3i)T^{2}
19 1+(−3.32−0.976i)T+(15.9+10.2i)T2 1 + (-3.32 - 0.976i)T + (15.9 + 10.2i)T^{2}
29 1+(2.24+7.65i)T+(−24.3+15.6i)T2 1 + (2.24 + 7.65i)T + (-24.3 + 15.6i)T^{2}
31 1+(−0.110+0.241i)T+(−20.3−23.4i)T2 1 + (-0.110 + 0.241i)T + (-20.3 - 23.4i)T^{2}
37 1+(0.0307+0.429i)T+(−36.6+5.26i)T2 1 + (0.0307 + 0.429i)T + (-36.6 + 5.26i)T^{2}
41 1+(2.05+2.37i)T+(−5.83+40.5i)T2 1 + (2.05 + 2.37i)T + (-5.83 + 40.5i)T^{2}
43 1+(−0.178+0.479i)T+(−32.4−28.1i)T2 1 + (-0.178 + 0.479i)T + (-32.4 - 28.1i)T^{2}
47 1+(4.99+4.99i)T+47iT2 1 + (4.99 + 4.99i)T + 47iT^{2}
53 1+(−2.17−9.99i)T+(−48.2+22.0i)T2 1 + (-2.17 - 9.99i)T + (-48.2 + 22.0i)T^{2}
59 1+(−0.591−0.919i)T+(−24.5+53.6i)T2 1 + (-0.591 - 0.919i)T + (-24.5 + 53.6i)T^{2}
61 1+(−0.500−0.228i)T+(39.9+46.1i)T2 1 + (-0.500 - 0.228i)T + (39.9 + 46.1i)T^{2}
67 1+(2.90+3.88i)T+(−18.8+64.2i)T2 1 + (2.90 + 3.88i)T + (-18.8 + 64.2i)T^{2}
71 1+(−2.16+15.0i)T+(−68.1−20.0i)T2 1 + (-2.16 + 15.0i)T + (-68.1 - 20.0i)T^{2}
73 1+(2.47−1.34i)T+(39.4−61.4i)T2 1 + (2.47 - 1.34i)T + (39.4 - 61.4i)T^{2}
79 1+(−5.55+3.56i)T+(32.8−71.8i)T2 1 + (-5.55 + 3.56i)T + (32.8 - 71.8i)T^{2}
83 1+(−17.4+1.24i)T+(82.1−11.8i)T2 1 + (-17.4 + 1.24i)T + (82.1 - 11.8i)T^{2}
89 1+(0.783+1.71i)T+(−58.2+67.2i)T2 1 + (0.783 + 1.71i)T + (-58.2 + 67.2i)T^{2}
97 1+(11.5+0.825i)T+(96.0+13.8i)T2 1 + (11.5 + 0.825i)T + (96.0 + 13.8i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.23300023179014943978852708873, −12.16388356501338256927634495078, −11.74062786678250799236576137735, −10.73320648861363331344106477355, −9.106075229499011169917698795235, −8.026582740209522445228759036154, −6.20670748606048602666917689966, −5.00233455899901569216451570717, −4.11348778612108461586032548663, −1.91025942024740598476435500280, 3.07175077250734540240709381559, 5.08118690429876746379530553757, 5.48367582589482470005013048154, 6.90639332131677385093260938550, 7.88469498050752204003249502482, 9.788543081824180716119628963296, 10.85617068285562044663935874850, 11.61575929742592371674328796146, 13.08962436604531071889886258612, 13.95303127272904610769022284651

Graph of the ZZ-function along the critical line