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Properties

Label 2-7e2-7.2-c9-0-12
Degree 22
Conductor 4949
Sign 0.605−0.795i0.605 - 0.795i
Analytic cond. 25.236725.2367
Root an. cond. 5.023615.02361
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)2-s + (243.5 + 421. i)4-s + 4.99e3·8-s + (9.84e3 − 1.70e4i)9-s + (2.14e4 + 3.71e4i)11-s + (−1.12e5 + 1.94e5i)16-s + (−4.92e4 − 8.52e4i)18-s + 2.14e5·22-s + (−6.98e5 + 1.20e6i)23-s + (9.76e5 + 1.69e6i)25-s + 7.57e6·29-s + (1.83e6 + 3.18e6i)32-s + 9.58e6·36-s + (−1.13e7 + 1.96e7i)37-s + 3.71e7·43-s + (−1.04e7 + 1.80e7i)44-s + ⋯
L(s)  = 1  + (0.110 − 0.191i)2-s + (0.475 + 0.823i)4-s + 0.431·8-s + (0.499 − 0.866i)9-s + (0.441 + 0.765i)11-s + (−0.427 + 0.741i)16-s + (−0.110 − 0.191i)18-s + 0.195·22-s + (−0.520 + 0.900i)23-s + (0.5 + 0.866i)25-s + 1.98·29-s + (0.310 + 0.537i)32-s + 0.951·36-s + (−0.997 + 1.72i)37-s + 1.65·43-s + (−0.420 + 0.727i)44-s + ⋯

Functional equation

Λ(s)=(49s/2 ΓC(s) L(s)=((0.605−0.795i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(49s/2 ΓC(s+9/2) L(s)=((0.605−0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4949    =    727^{2}
Sign: 0.605−0.795i0.605 - 0.795i
Analytic conductor: 25.236725.2367
Root analytic conductor: 5.023615.02361
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ49(30,⋅)\chi_{49} (30, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 49, ( :9/2), 0.605−0.795i)(2,\ 49,\ (\ :9/2),\ 0.605 - 0.795i)

Particular Values

L(5)L(5) ≈\approx 2.17601+1.07871i2.17601 + 1.07871i
L(12)L(\frac12) ≈\approx 2.17601+1.07871i2.17601 + 1.07871i
L(112)L(\frac{11}{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)
bad7 1 1
good2 1+(−2.5+4.33i)T+(−256−443.i)T2 1 + (-2.5 + 4.33i)T + (-256 - 443. i)T^{2}
3 1+(−9.84e3+1.70e4i)T2 1 + (-9.84e3 + 1.70e4i)T^{2}
5 1+(−9.76e5−1.69e6i)T2 1 + (-9.76e5 - 1.69e6i)T^{2}
11 1+(−2.14e4−3.71e4i)T+(−1.17e9+2.04e9i)T2 1 + (-2.14e4 - 3.71e4i)T + (-1.17e9 + 2.04e9i)T^{2}
13 1+1.06e10T2 1 + 1.06e10T^{2}
17 1+(−5.92e10+1.02e11i)T2 1 + (-5.92e10 + 1.02e11i)T^{2}
19 1+(−1.61e11−2.79e11i)T2 1 + (-1.61e11 - 2.79e11i)T^{2}
23 1+(6.98e5−1.20e6i)T+(−9.00e11−1.55e12i)T2 1 + (6.98e5 - 1.20e6i)T + (-9.00e11 - 1.55e12i)T^{2}
29 1−7.57e6T+1.45e13T2 1 - 7.57e6T + 1.45e13T^{2}
31 1+(−1.32e13+2.28e13i)T2 1 + (-1.32e13 + 2.28e13i)T^{2}
37 1+(1.13e7−1.96e7i)T+(−6.49e13−1.12e14i)T2 1 + (1.13e7 - 1.96e7i)T + (-6.49e13 - 1.12e14i)T^{2}
41 1+3.27e14T2 1 + 3.27e14T^{2}
43 1−3.71e7T+5.02e14T2 1 - 3.71e7T + 5.02e14T^{2}
47 1+(−5.59e14−9.69e14i)T2 1 + (-5.59e14 - 9.69e14i)T^{2}
53 1+(−2.90e7−5.03e7i)T+(−1.64e15+2.85e15i)T2 1 + (-2.90e7 - 5.03e7i)T + (-1.64e15 + 2.85e15i)T^{2}
59 1+(−4.33e15+7.50e15i)T2 1 + (-4.33e15 + 7.50e15i)T^{2}
61 1+(−5.84e15−1.01e16i)T2 1 + (-5.84e15 - 1.01e16i)T^{2}
67 1+(1.31e8+2.27e8i)T+(−1.36e16+2.35e16i)T2 1 + (1.31e8 + 2.27e8i)T + (-1.36e16 + 2.35e16i)T^{2}
71 1+4.13e8T+4.58e16T2 1 + 4.13e8T + 4.58e16T^{2}
73 1+(−2.94e16+5.09e16i)T2 1 + (-2.94e16 + 5.09e16i)T^{2}
79 1+(−3.01e8+5.22e8i)T+(−5.99e16−1.03e17i)T2 1 + (-3.01e8 + 5.22e8i)T + (-5.99e16 - 1.03e17i)T^{2}
83 1+1.86e17T2 1 + 1.86e17T^{2}
89 1+(−1.75e17−3.03e17i)T2 1 + (-1.75e17 - 3.03e17i)T^{2}
97 1+7.60e17T2 1 + 7.60e17T^{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

−13.63710372481755046855608412281, −12.39650430104034190810895247771, −11.82276801258902823245907127143, −10.32258873166641244426943656185, −8.997213593298654191809275746383, −7.51427255124369556147385542311, −6.49776454154230393379984601939, −4.42011980977517455192126043811, −3.14068781967076920257556481988, −1.44800155633274376186333954699, 0.851354735513381158997470362782, 2.37409109526516569093004363661, 4.49096994358657966389670122647, 5.87637176174347950077153352495, 7.05305963664313701613200405744, 8.548203908787923868595947359097, 10.16786960097409010593433827994, 10.91101440218544736316144532331, 12.29108240145796027081282825788, 13.80794100938654615419601184942

Graph of the ZZ-function along the critical line