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Properties

Label 2-115-115.114-c4-0-15
Degree 22
Conductor 115115
Sign −0.872−0.489i-0.872 - 0.489i
Analytic cond. 11.887511.8875
Root an. cond. 3.447833.44783
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.17i·2-s − 1.68i·3-s − 10.7·4-s + (−10.7 + 22.5i)5-s + 8.72·6-s + 91.0·7-s + 26.9i·8-s + 78.1·9-s + (−116. − 55.7i)10-s + 122. i·11-s + 18.1i·12-s − 261. i·13-s + 471. i·14-s + (38.0 + 18.1i)15-s − 312.·16-s − 361.·17-s + ⋯
L(s)  = 1  + 1.29i·2-s − 0.187i·3-s − 0.674·4-s + (−0.430 + 0.902i)5-s + 0.242·6-s + 1.85·7-s + 0.421i·8-s + 0.964·9-s + (−1.16 − 0.557i)10-s + 1.01i·11-s + 0.126i·12-s − 1.54i·13-s + 2.40i·14-s + (0.169 + 0.0807i)15-s − 1.21·16-s − 1.25·17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.872−0.489i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+2) L(s)=((−0.872−0.489i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.872−0.489i-0.872 - 0.489i
Analytic conductor: 11.887511.8875
Root analytic conductor: 3.447833.44783
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ115(114,⋅)\chi_{115} (114, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :2), −0.872−0.489i)(2,\ 115,\ (\ :2),\ -0.872 - 0.489i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.505944+1.93605i0.505944 + 1.93605i
L(12)L(\frac12) ≈\approx 0.505944+1.93605i0.505944 + 1.93605i
L(3)L(3) 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+(10.7−22.5i)T 1 + (10.7 - 22.5i)T
23 1+(34.8−527.i)T 1 + (34.8 - 527. i)T
good2 1−5.17iT−16T2 1 - 5.17iT - 16T^{2}
3 1+1.68iT−81T2 1 + 1.68iT - 81T^{2}
7 1−91.0T+2.40e3T2 1 - 91.0T + 2.40e3T^{2}
11 1−122.iT−1.46e4T2 1 - 122. iT - 1.46e4T^{2}
13 1+261.iT−2.85e4T2 1 + 261. iT - 2.85e4T^{2}
17 1+361.T+8.35e4T2 1 + 361.T + 8.35e4T^{2}
19 1−477.iT−1.30e5T2 1 - 477. iT - 1.30e5T^{2}
29 1−985.T+7.07e5T2 1 - 985.T + 7.07e5T^{2}
31 1+675.T+9.23e5T2 1 + 675.T + 9.23e5T^{2}
37 1+230.T+1.87e6T2 1 + 230.T + 1.87e6T^{2}
41 1+1.08e3T+2.82e6T2 1 + 1.08e3T + 2.82e6T^{2}
43 1−2.03e3T+3.41e6T2 1 - 2.03e3T + 3.41e6T^{2}
47 1+2.21e3iT−4.87e6T2 1 + 2.21e3iT - 4.87e6T^{2}
53 1−745.T+7.89e6T2 1 - 745.T + 7.89e6T^{2}
59 1−703.T+1.21e7T2 1 - 703.T + 1.21e7T^{2}
61 1+570.iT−1.38e7T2 1 + 570. iT - 1.38e7T^{2}
67 1−3.64e3T+2.01e7T2 1 - 3.64e3T + 2.01e7T^{2}
71 1+759.T+2.54e7T2 1 + 759.T + 2.54e7T^{2}
73 1+879.iT−2.83e7T2 1 + 879. iT - 2.83e7T^{2}
79 1−2.23e3iT−3.89e7T2 1 - 2.23e3iT - 3.89e7T^{2}
83 1+6.08e3T+4.74e7T2 1 + 6.08e3T + 4.74e7T^{2}
89 1+25.1iT−6.27e7T2 1 + 25.1iT - 6.27e7T^{2}
97 1−3.90e3T+8.85e7T2 1 - 3.90e3T + 8.85e7T^{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

−13.71805045671232074913203701260, −12.26448454959736993396165109429, −11.14146371932991866710278703436, −10.22804501188276941226043784661, −8.303064045805328696745860474094, −7.66490326771781680403555931683, −6.96104568633496713323925995509, −5.47339191184111550358861639071, −4.32010027486929037666503373345, −1.95678391717539233397637606784, 0.952633448465848130392419319017, 2.08093323978243459159484124303, 4.27972231482974966965672433579, 4.66007370833379695372545306486, 6.96766908657424250266543717358, 8.525170094290322457756345358508, 9.188066869551593832281757144895, 10.82633078692789077200067836771, 11.30166642576068607443426694313, 12.09797923469213446168024790246

Graph of the ZZ-function along the critical line