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Properties

Label 2-115-115.114-c4-0-5
Degree 22
Conductor 115115
Sign −0.990+0.134i-0.990 + 0.134i
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  + 6.07i·2-s − 5.74i·3-s − 20.9·4-s + (19.8 + 15.2i)5-s + 34.9·6-s − 44.4·7-s − 29.8i·8-s + 47.9·9-s + (−92.6 + 120. i)10-s + 28.8i·11-s + 120. i·12-s + 250. i·13-s − 269. i·14-s + (87.6 − 113. i)15-s − 153.·16-s − 450.·17-s + ⋯
L(s)  = 1  + 1.51i·2-s − 0.638i·3-s − 1.30·4-s + (0.792 + 0.609i)5-s + 0.970·6-s − 0.906·7-s − 0.465i·8-s + 0.592·9-s + (−0.926 + 1.20i)10-s + 0.238i·11-s + 0.834i·12-s + 1.48i·13-s − 1.37i·14-s + (0.389 − 0.506i)15-s − 0.599·16-s − 1.55·17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((−0.990+0.134i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+2) L(s)=((−0.990+0.134i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: −0.990+0.134i-0.990 + 0.134i
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), −0.990+0.134i)(2,\ 115,\ (\ :2),\ -0.990 + 0.134i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.0900503−1.33222i0.0900503 - 1.33222i
L(12)L(\frac12) ≈\approx 0.0900503−1.33222i0.0900503 - 1.33222i
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+(−19.8−15.2i)T 1 + (-19.8 - 15.2i)T
23 1+(371.−376.i)T 1 + (371. - 376. i)T
good2 1−6.07iT−16T2 1 - 6.07iT - 16T^{2}
3 1+5.74iT−81T2 1 + 5.74iT - 81T^{2}
7 1+44.4T+2.40e3T2 1 + 44.4T + 2.40e3T^{2}
11 1−28.8iT−1.46e4T2 1 - 28.8iT - 1.46e4T^{2}
13 1−250.iT−2.85e4T2 1 - 250. iT - 2.85e4T^{2}
17 1+450.T+8.35e4T2 1 + 450.T + 8.35e4T^{2}
19 1+109.iT−1.30e5T2 1 + 109. iT - 1.30e5T^{2}
29 1+103.T+7.07e5T2 1 + 103.T + 7.07e5T^{2}
31 1−874.T+9.23e5T2 1 - 874.T + 9.23e5T^{2}
37 1+494.T+1.87e6T2 1 + 494.T + 1.87e6T^{2}
41 1−563.T+2.82e6T2 1 - 563.T + 2.82e6T^{2}
43 1−1.72e3T+3.41e6T2 1 - 1.72e3T + 3.41e6T^{2}
47 1−3.28e3iT−4.87e6T2 1 - 3.28e3iT - 4.87e6T^{2}
53 1+4.26e3T+7.89e6T2 1 + 4.26e3T + 7.89e6T^{2}
59 1−4.67e3T+1.21e7T2 1 - 4.67e3T + 1.21e7T^{2}
61 1−6.20e3iT−1.38e7T2 1 - 6.20e3iT - 1.38e7T^{2}
67 1−6.04e3T+2.01e7T2 1 - 6.04e3T + 2.01e7T^{2}
71 1−2.62e3T+2.54e7T2 1 - 2.62e3T + 2.54e7T^{2}
73 1+1.55e3iT−2.83e7T2 1 + 1.55e3iT - 2.83e7T^{2}
79 1+1.54e3iT−3.89e7T2 1 + 1.54e3iT - 3.89e7T^{2}
83 1−1.05e3T+4.74e7T2 1 - 1.05e3T + 4.74e7T^{2}
89 1+1.18e4iT−6.27e7T2 1 + 1.18e4iT - 6.27e7T^{2}
97 1−8.50e3T+8.85e7T2 1 - 8.50e3T + 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

−13.64578760011755261850201018966, −12.90080597633787117615917114595, −11.36152654838356503129874362738, −9.796637997145844505398386491361, −8.993939254463844635689705371937, −7.42237137580842385717786705275, −6.65194869481599019865674129282, −6.18478437432020025029897192591, −4.44177840194190371230429409842, −2.13907854265670022084796164582, 0.56170851346014634904512432419, 2.28363952782620619550988933942, 3.65854045441226126845432523157, 4.88113228655595901365720054503, 6.42979661828591897663180102641, 8.567215904987235142566191167746, 9.672110966201181969956011691587, 10.14439401655419822618571245816, 11.01736908359041916640290669916, 12.59030671349804963371198095925

Graph of the ZZ-function along the critical line