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Properties

Label 2-51-3.2-c6-0-9
Degree 22
Conductor 5151
Sign −0.00655−0.999i-0.00655 - 0.999i
Analytic cond. 11.732711.7327
Root an. cond. 3.425313.42531
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.98i·2-s + (−26.9 + 0.177i)3-s − 16.6·4-s − 127. i·5-s + (−1.59 − 242. i)6-s + 363.·7-s + 424. i·8-s + (728. − 9.56i)9-s + 1.14e3·10-s + 522. i·11-s + (450. − 2.95i)12-s − 1.11e3·13-s + 3.26e3i·14-s + (22.5 + 3.43e3i)15-s − 4.88e3·16-s − 1.19e3i·17-s + ⋯
L(s)  = 1  + 1.12i·2-s + (−0.999 + 0.00655i)3-s − 0.260·4-s − 1.01i·5-s + (−0.00736 − 1.12i)6-s + 1.05·7-s + 0.830i·8-s + (0.999 − 0.0131i)9-s + 1.14·10-s + 0.392i·11-s + (0.260 − 0.00171i)12-s − 0.507·13-s + 1.18i·14-s + (0.00667 + 1.01i)15-s − 1.19·16-s − 0.242i·17-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=((−0.00655−0.999i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00655 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+3) L(s)=((−0.00655−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.00655 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: −0.00655−0.999i-0.00655 - 0.999i
Analytic conductor: 11.732711.7327
Root analytic conductor: 3.425313.42531
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ51(35,⋅)\chi_{51} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 51, ( :3), −0.00655−0.999i)(2,\ 51,\ (\ :3),\ -0.00655 - 0.999i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.08581+1.09295i1.08581 + 1.09295i
L(12)L(\frac12) ≈\approx 1.08581+1.09295i1.08581 + 1.09295i
L(4)L(4) 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)
bad3 1+(26.9−0.177i)T 1 + (26.9 - 0.177i)T
17 1+1.19e3iT 1 + 1.19e3iT
good2 1−8.98iT−64T2 1 - 8.98iT - 64T^{2}
5 1+127.iT−1.56e4T2 1 + 127. iT - 1.56e4T^{2}
7 1−363.T+1.17e5T2 1 - 363.T + 1.17e5T^{2}
11 1−522.iT−1.77e6T2 1 - 522. iT - 1.77e6T^{2}
13 1+1.11e3T+4.82e6T2 1 + 1.11e3T + 4.82e6T^{2}
19 1−1.20e4T+4.70e7T2 1 - 1.20e4T + 4.70e7T^{2}
23 1−1.94e4iT−1.48e8T2 1 - 1.94e4iT - 1.48e8T^{2}
29 1−3.50e4iT−5.94e8T2 1 - 3.50e4iT - 5.94e8T^{2}
31 1−4.16e4T+8.87e8T2 1 - 4.16e4T + 8.87e8T^{2}
37 1−6.10e4T+2.56e9T2 1 - 6.10e4T + 2.56e9T^{2}
41 1−3.51e4iT−4.75e9T2 1 - 3.51e4iT - 4.75e9T^{2}
43 1+7.71e3T+6.32e9T2 1 + 7.71e3T + 6.32e9T^{2}
47 1+1.54e5iT−1.07e10T2 1 + 1.54e5iT - 1.07e10T^{2}
53 1+1.29e5iT−2.21e10T2 1 + 1.29e5iT - 2.21e10T^{2}
59 1+1.36e5iT−4.21e10T2 1 + 1.36e5iT - 4.21e10T^{2}
61 1−1.01e5T+5.15e10T2 1 - 1.01e5T + 5.15e10T^{2}
67 1+2.07e5T+9.04e10T2 1 + 2.07e5T + 9.04e10T^{2}
71 1−2.38e5iT−1.28e11T2 1 - 2.38e5iT - 1.28e11T^{2}
73 1+6.89e4T+1.51e11T2 1 + 6.89e4T + 1.51e11T^{2}
79 1−5.54e5T+2.43e11T2 1 - 5.54e5T + 2.43e11T^{2}
83 1−1.27e5iT−3.26e11T2 1 - 1.27e5iT - 3.26e11T^{2}
89 1−4.25e5iT−4.96e11T2 1 - 4.25e5iT - 4.96e11T^{2}
97 1+6.43e5T+8.32e11T2 1 + 6.43e5T + 8.32e11T^{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

−14.85325518150565907101474130827, −13.54144337769919267032850466813, −12.04962628806736170988398953051, −11.34382416764094291720712398056, −9.594206268465959258352493799695, −8.038354516314086845291990985253, −7.06207146630541498631505390481, −5.36583878586900592977379562313, −4.89983020117899784172881582158, −1.30183482423834703134902656737, 0.943762358154642334067200957573, 2.65724525353442648246551884537, 4.50068425430988643042288256191, 6.27028431197967888037590653965, 7.59586033967090571683767791373, 9.848174905164797301336526345995, 10.76874179619108534075231939744, 11.47872674829608514716136370226, 12.24969975190222163315098826971, 13.76614104342984156267245745005

Graph of the ZZ-function along the critical line