Properties

Label 1-95-95.17-r1-0-0
Degree 11
Conductor 9595
Sign 0.9820.187i-0.982 - 0.187i
Analytic cond. 10.209110.2091
Root an. cond. 10.209110.2091
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  + (0.342 − 0.939i)2-s + (0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.173 − 0.984i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.984 − 0.173i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s i·18-s + (−0.939 − 0.342i)21-s + (−0.984 + 0.173i)22-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.984 − 0.173i)3-s + (−0.766 − 0.642i)4-s + (0.173 − 0.984i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.939 − 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.984 − 0.173i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s i·18-s + (−0.939 − 0.342i)21-s + (−0.984 + 0.173i)22-s + ⋯

Functional equation

Λ(s)=(95s/2ΓR(s+1)L(s)=((0.9820.187i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(95s/2ΓR(s+1)L(s)=((0.9820.187i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 9595    =    5195 \cdot 19
Sign: 0.9820.187i-0.982 - 0.187i
Analytic conductor: 10.209110.2091
Root analytic conductor: 10.209110.2091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ95(17,)\chi_{95} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 95, (1: ), 0.9820.187i)(1,\ 95,\ (1:\ ),\ -0.982 - 0.187i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.17678021701.868072433i0.1767802170 - 1.868072433i
L(12)L(\frac12) \approx 0.17678021701.868072433i0.1767802170 - 1.868072433i
L(1)L(1) \approx 0.92235658610.9931757242i0.9223565861 - 0.9931757242i
L(1)L(1) \approx 0.92235658610.9931757242i0.9223565861 - 0.9931757242i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
19 1 1
good2 1+(0.3420.939i)T 1 + (0.342 - 0.939i)T
3 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
7 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
11 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
13 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
17 1+(0.3420.939i)T 1 + (0.342 - 0.939i)T
23 1+(0.6420.766i)T 1 + (0.642 - 0.766i)T
29 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
31 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
37 1+iT 1 + iT
41 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
43 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
47 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
53 1+(0.6420.766i)T 1 + (0.642 - 0.766i)T
59 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
61 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
67 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
71 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
73 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
79 1+(0.1730.984i)T 1 + (-0.173 - 0.984i)T
83 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
89 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
97 1+(0.3420.939i)T 1 + (0.342 - 0.939i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−30.98135770518217531849514316454, −29.62812808914594419659100819449, −28.09104119273540869789990167435, −26.91429010043483726294250617971, −25.92172520323100338441854735468, −25.407476203211366458521523823429, −24.37425853069745485940614474263, −23.19992730323075993973255197939, −22.01392563004468109425444409266, −21.19720271091912799640121382338, −19.71348438815784601881966168497, −18.76154287650181773569324831111, −17.418043975058359146609800650165, −16.10412777332303770928350032251, −15.22478334540377254566419336291, −14.46433164496284062734975852764, −13.087480482507805945294002787446, −12.44098821233523486340195659076, −9.97829340533415089742588057409, −9.1514200243059098579517436602, −7.86017759675165875025532754001, −6.86568591240963037166991393461, −5.27218632101250518940891513829, −3.89119442824375811129816938206, −2.55597237984191897337141684415, 0.66769473517189660564445617831, 2.61136917444070395285106627891, 3.41231115360505481604069947558, 4.948119818026468540083078003138, 6.80971678393313520431257633691, 8.375271067209906133229225270198, 9.61021203171702628485831432351, 10.43667545784786827476793935019, 12.08531733577547896342186510867, 13.165931551949986314580798249970, 13.88806909531061409519570712478, 15.007569792172347794277779056984, 16.4244020102644219776892076580, 18.198270858395307907851027038190, 19.16426091763895587643733981010, 19.86815435623128000085061756300, 20.84607498271861998889276216975, 21.85294329714583465939167076019, 23.018794871997052769039860065100, 24.1190196008869760289567351059, 25.26897192213336807253471232055, 26.70738251597767160050846473916, 27.07796898200707536148837042582, 28.90450713008216150070872378823, 29.48330036745819583981651496752

Graph of the ZZ-function along the critical line