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Properties

Label 2-115-115.114-c4-0-23
Degree 22
Conductor 115115
Sign 0.880+0.473i0.880 + 0.473i
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  − 1.05i·2-s − 9.44i·3-s + 14.8·4-s + (−2.52 + 24.8i)5-s − 10.0·6-s + 67.6·7-s − 32.7i·8-s − 8.12·9-s + (26.3 + 2.67i)10-s + 168. i·11-s − 140. i·12-s + 214. i·13-s − 71.7i·14-s + (234. + 23.8i)15-s + 203.·16-s − 13.4·17-s + ⋯
L(s)  = 1  − 0.264i·2-s − 1.04i·3-s + 0.929·4-s + (−0.101 + 0.994i)5-s − 0.277·6-s + 1.38·7-s − 0.511i·8-s − 0.100·9-s + (0.263 + 0.0267i)10-s + 1.38i·11-s − 0.975i·12-s + 1.27i·13-s − 0.366i·14-s + (1.04 + 0.106i)15-s + 0.794·16-s − 0.0463·17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.880+0.473i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+2) L(s)=((0.880+0.473i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.880+0.473i0.880 + 0.473i
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), 0.880+0.473i)(2,\ 115,\ (\ :2),\ 0.880 + 0.473i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.43938−0.614539i2.43938 - 0.614539i
L(12)L(\frac12) ≈\approx 2.43938−0.614539i2.43938 - 0.614539i
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+(2.52−24.8i)T 1 + (2.52 - 24.8i)T
23 1+(−202.+488.i)T 1 + (-202. + 488. i)T
good2 1+1.05iT−16T2 1 + 1.05iT - 16T^{2}
3 1+9.44iT−81T2 1 + 9.44iT - 81T^{2}
7 1−67.6T+2.40e3T2 1 - 67.6T + 2.40e3T^{2}
11 1−168.iT−1.46e4T2 1 - 168. iT - 1.46e4T^{2}
13 1−214.iT−2.85e4T2 1 - 214. iT - 2.85e4T^{2}
17 1+13.4T+8.35e4T2 1 + 13.4T + 8.35e4T^{2}
19 1+362.iT−1.30e5T2 1 + 362. iT - 1.30e5T^{2}
29 1+1.21e3T+7.07e5T2 1 + 1.21e3T + 7.07e5T^{2}
31 1−1.39e3T+9.23e5T2 1 - 1.39e3T + 9.23e5T^{2}
37 1−1.06e3T+1.87e6T2 1 - 1.06e3T + 1.87e6T^{2}
41 1+541.T+2.82e6T2 1 + 541.T + 2.82e6T^{2}
43 1+2.34e3T+3.41e6T2 1 + 2.34e3T + 3.41e6T^{2}
47 1+416.iT−4.87e6T2 1 + 416. iT - 4.87e6T^{2}
53 1+4.14e3T+7.89e6T2 1 + 4.14e3T + 7.89e6T^{2}
59 1+857.T+1.21e7T2 1 + 857.T + 1.21e7T^{2}
61 1−3.04e3iT−1.38e7T2 1 - 3.04e3iT - 1.38e7T^{2}
67 1−1.87e3T+2.01e7T2 1 - 1.87e3T + 2.01e7T^{2}
71 1−9.87e3T+2.54e7T2 1 - 9.87e3T + 2.54e7T^{2}
73 1+6.97e3iT−2.83e7T2 1 + 6.97e3iT - 2.83e7T^{2}
79 1−2.45e3iT−3.89e7T2 1 - 2.45e3iT - 3.89e7T^{2}
83 1+1.22e4T+4.74e7T2 1 + 1.22e4T + 4.74e7T^{2}
89 1−8.56e3iT−6.27e7T2 1 - 8.56e3iT - 6.27e7T^{2}
97 1+8.34e3T+8.85e7T2 1 + 8.34e3T + 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

−12.53413534658534197989560329839, −11.62035444018891115821290601418, −11.09331811608325465646965185137, −9.820890911895211761617658541483, −7.975250239775052126666462585819, −7.08546417659202416938922684441, −6.58651165962830825798898462697, −4.50622301904416835592357788570, −2.37766849972395423804695269991, −1.63445498780254574499334687613, 1.35402621710588938524822371876, 3.49150981616805223685491383356, 5.01533817897392517263168965137, 5.76987835805759309610500450082, 7.84289270078122439264555354135, 8.367603603728335396409001805579, 9.849174229922201427071730840406, 11.00325728075792682845810927613, 11.53914227777214197841171266298, 12.89147645510807822839615646509

Graph of the ZZ-function along the critical line