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Properties

Label 2-1250-1.1-c3-0-75
Degree 22
Conductor 12501250
Sign 11
Analytic cond. 73.752373.7523
Root an. cond. 8.587928.58792
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 8.30·3-s + 4·4-s − 16.6·6-s + 11.9·7-s − 8·8-s + 41.9·9-s + 54.3·11-s + 33.2·12-s + 67.8·13-s − 23.9·14-s + 16·16-s + 10.6·17-s − 83.9·18-s + 22.9·19-s + 99.3·21-s − 108.·22-s + 147.·23-s − 66.4·24-s − 135.·26-s + 124.·27-s + 47.8·28-s + 205.·29-s − 87.6·31-s − 32·32-s + 451.·33-s − 21.2·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.59·3-s + 0.5·4-s − 1.13·6-s + 0.645·7-s − 0.353·8-s + 1.55·9-s + 1.48·11-s + 0.799·12-s + 1.44·13-s − 0.456·14-s + 0.250·16-s + 0.151·17-s − 1.09·18-s + 0.277·19-s + 1.03·21-s − 1.05·22-s + 1.33·23-s − 0.565·24-s − 1.02·26-s + 0.886·27-s + 0.322·28-s + 1.31·29-s − 0.507·31-s − 0.176·32-s + 2.38·33-s − 0.107·34-s + ⋯

Functional equation

Λ(s)=(1250s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1250s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12501250    =    2⋅542 \cdot 5^{4}
Sign: 11
Analytic conductor: 73.752373.7523
Root analytic conductor: 8.587928.58792
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1250, ( :3/2), 1)(2,\ 1250,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.9339685793.933968579
L(12)L(\frac12) ≈\approx 3.9339685793.933968579
L(52)L(\frac{5}{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)
bad2 1+2T 1 + 2T
5 1 1
good3 1−8.30T+27T2 1 - 8.30T + 27T^{2}
7 1−11.9T+343T2 1 - 11.9T + 343T^{2}
11 1−54.3T+1.33e3T2 1 - 54.3T + 1.33e3T^{2}
13 1−67.8T+2.19e3T2 1 - 67.8T + 2.19e3T^{2}
17 1−10.6T+4.91e3T2 1 - 10.6T + 4.91e3T^{2}
19 1−22.9T+6.85e3T2 1 - 22.9T + 6.85e3T^{2}
23 1−147.T+1.21e4T2 1 - 147.T + 1.21e4T^{2}
29 1−205.T+2.43e4T2 1 - 205.T + 2.43e4T^{2}
31 1+87.6T+2.97e4T2 1 + 87.6T + 2.97e4T^{2}
37 1+413.T+5.06e4T2 1 + 413.T + 5.06e4T^{2}
41 1+67.6T+6.89e4T2 1 + 67.6T + 6.89e4T^{2}
43 1+471.T+7.95e4T2 1 + 471.T + 7.95e4T^{2}
47 1+66.0T+1.03e5T2 1 + 66.0T + 1.03e5T^{2}
53 1+316.T+1.48e5T2 1 + 316.T + 1.48e5T^{2}
59 1+101.T+2.05e5T2 1 + 101.T + 2.05e5T^{2}
61 1+405.T+2.26e5T2 1 + 405.T + 2.26e5T^{2}
67 1−525.T+3.00e5T2 1 - 525.T + 3.00e5T^{2}
71 1+477.T+3.57e5T2 1 + 477.T + 3.57e5T^{2}
73 1−857.T+3.89e5T2 1 - 857.T + 3.89e5T^{2}
79 1−549.T+4.93e5T2 1 - 549.T + 4.93e5T^{2}
83 1+287.T+5.71e5T2 1 + 287.T + 5.71e5T^{2}
89 1+39.9T+7.04e5T2 1 + 39.9T + 7.04e5T^{2}
97 1−1.02e3T+9.12e5T2 1 - 1.02e3T + 9.12e5T^{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

−9.035498750786148777313786603501, −8.610055840474365079302271785076, −8.078936411383808078814557644818, −7.03963551977920677093940365914, −6.41642167248270792478702351879, −4.92139498869641512839693129178, −3.68033320470698989089948981847, −3.17331508518770915275983494402, −1.74736175494083124537429285420, −1.21844710552070063501257751982, 1.21844710552070063501257751982, 1.74736175494083124537429285420, 3.17331508518770915275983494402, 3.68033320470698989089948981847, 4.92139498869641512839693129178, 6.41642167248270792478702351879, 7.03963551977920677093940365914, 8.078936411383808078814557644818, 8.610055840474365079302271785076, 9.035498750786148777313786603501

Graph of the ZZ-function along the critical line