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Properties

Label 2-59-59.10-c6-0-2
Degree 22
Conductor 5959
Sign −0.996+0.0841i-0.996 + 0.0841i
Analytic cond. 13.573113.5731
Root an. cond. 3.684183.68418
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  + (3.80 + 1.51i)2-s + (26.7 + 2.91i)3-s + (−34.2 − 32.4i)4-s + (−125. + 148. i)5-s + (97.5 + 51.7i)6-s + (−29.7 − 17.9i)7-s + (−191. − 413. i)8-s + (−2.40 − 0.529i)9-s + (−704. + 373. i)10-s + (−1.82e3 − 99.2i)11-s + (−823. − 970. i)12-s + (−176. − 804. i)13-s + (−86.1 − 113. i)14-s + (−3.80e3 + 3.60e3i)15-s + (62.5 + 1.15e3i)16-s + (2.32e3 − 1.40e3i)17-s + ⋯
L(s)  = 1  + (0.475 + 0.189i)2-s + (0.992 + 0.107i)3-s + (−0.535 − 0.507i)4-s + (−1.00 + 1.18i)5-s + (0.451 + 0.239i)6-s + (−0.0868 − 0.0522i)7-s + (−0.373 − 0.807i)8-s + (−0.00329 − 0.000726i)9-s + (−0.704 + 0.373i)10-s + (−1.37 − 0.0745i)11-s + (−0.476 − 0.561i)12-s + (−0.0805 − 0.365i)13-s + (−0.0313 − 0.0412i)14-s + (−1.12 + 1.06i)15-s + (0.0152 + 0.281i)16-s + (0.473 − 0.284i)17-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=((−0.996+0.0841i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=((−0.996+0.0841i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: −0.996+0.0841i-0.996 + 0.0841i
Analytic conductor: 13.573113.5731
Root analytic conductor: 3.684183.68418
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ59(10,⋅)\chi_{59} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 59, ( :3), −0.996+0.0841i)(2,\ 59,\ (\ :3),\ -0.996 + 0.0841i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.0120787−0.286600i0.0120787 - 0.286600i
L(12)L(\frac12) ≈\approx 0.0120787−0.286600i0.0120787 - 0.286600i
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)
bad59 1+(−7.82e4−1.89e5i)T 1 + (-7.82e4 - 1.89e5i)T
good2 1+(−3.80−1.51i)T+(46.4+44.0i)T2 1 + (-3.80 - 1.51i)T + (46.4 + 44.0i)T^{2}
3 1+(−26.7−2.91i)T+(711.+156.i)T2 1 + (-26.7 - 2.91i)T + (711. + 156. i)T^{2}
5 1+(125.−148.i)T+(−2.52e3−1.54e4i)T2 1 + (125. - 148. i)T + (-2.52e3 - 1.54e4i)T^{2}
7 1+(29.7+17.9i)T+(5.51e4+1.03e5i)T2 1 + (29.7 + 17.9i)T + (5.51e4 + 1.03e5i)T^{2}
11 1+(1.82e3+99.2i)T+(1.76e6+1.91e5i)T2 1 + (1.82e3 + 99.2i)T + (1.76e6 + 1.91e5i)T^{2}
13 1+(176.+804.i)T+(−4.38e6+2.02e6i)T2 1 + (176. + 804. i)T + (-4.38e6 + 2.02e6i)T^{2}
17 1+(−2.32e3+1.40e3i)T+(1.13e7−2.13e7i)T2 1 + (-2.32e3 + 1.40e3i)T + (1.13e7 - 2.13e7i)T^{2}
19 1+(1.77e3−6.37e3i)T+(−4.03e7−2.42e7i)T2 1 + (1.77e3 - 6.37e3i)T + (-4.03e7 - 2.42e7i)T^{2}
23 1+(5.26e3−3.56e3i)T+(5.47e7−1.37e8i)T2 1 + (5.26e3 - 3.56e3i)T + (5.47e7 - 1.37e8i)T^{2}
29 1+(−6.05e3−1.51e4i)T+(−4.31e8+4.09e8i)T2 1 + (-6.05e3 - 1.51e4i)T + (-4.31e8 + 4.09e8i)T^{2}
31 1+(1.02e4−2.85e3i)T+(7.60e8−4.57e8i)T2 1 + (1.02e4 - 2.85e3i)T + (7.60e8 - 4.57e8i)T^{2}
37 1+(−4.05e3+8.76e3i)T+(−1.66e9−1.95e9i)T2 1 + (-4.05e3 + 8.76e3i)T + (-1.66e9 - 1.95e9i)T^{2}
41 1+(−2.28e4+3.37e4i)T+(−1.75e9−4.41e9i)T2 1 + (-2.28e4 + 3.37e4i)T + (-1.75e9 - 4.41e9i)T^{2}
43 1+(−6.41e4+3.47e3i)T+(6.28e9−6.83e8i)T2 1 + (-6.41e4 + 3.47e3i)T + (6.28e9 - 6.83e8i)T^{2}
47 1+(−1.70e4+1.44e4i)T+(1.74e9−1.06e10i)T2 1 + (-1.70e4 + 1.44e4i)T + (1.74e9 - 1.06e10i)T^{2}
53 1+(9.43e4−1.77e5i)T+(−1.24e10−1.83e10i)T2 1 + (9.43e4 - 1.77e5i)T + (-1.24e10 - 1.83e10i)T^{2}
61 1+(−1.64e5−6.54e4i)T+(3.74e10+3.54e10i)T2 1 + (-1.64e5 - 6.54e4i)T + (3.74e10 + 3.54e10i)T^{2}
67 1+(1.38e5+3.00e5i)T+(−5.85e10+6.89e10i)T2 1 + (1.38e5 + 3.00e5i)T + (-5.85e10 + 6.89e10i)T^{2}
71 1+(−3.03e5−3.57e5i)T+(−2.07e10+1.26e11i)T2 1 + (-3.03e5 - 3.57e5i)T + (-2.07e10 + 1.26e11i)T^{2}
73 1+(−9.31e4−1.22e5i)T+(−4.04e10+1.45e11i)T2 1 + (-9.31e4 - 1.22e5i)T + (-4.04e10 + 1.45e11i)T^{2}
79 1+(1.66e5−1.81e4i)T+(2.37e11−5.22e10i)T2 1 + (1.66e5 - 1.81e4i)T + (2.37e11 - 5.22e10i)T^{2}
83 1+(2.77e5+8.24e5i)T+(−2.60e11+1.97e11i)T2 1 + (2.77e5 + 8.24e5i)T + (-2.60e11 + 1.97e11i)T^{2}
89 1+(9.39e5−3.74e5i)T+(3.60e11−3.41e11i)T2 1 + (9.39e5 - 3.74e5i)T + (3.60e11 - 3.41e11i)T^{2}
97 1+(6.93e5−9.12e5i)T+(−2.22e11−8.02e11i)T2 1 + (6.93e5 - 9.12e5i)T + (-2.22e11 - 8.02e11i)T^{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

−14.51412158167275405989269350421, −13.74831348616399756010787941318, −12.46562599330831346547119562555, −10.85444433128089811330641563611, −9.900101422500633841378818028640, −8.350812032342961203767352745648, −7.37218448128994847123508485946, −5.65937799827426863587510132785, −3.87244146543973356310654583697, −2.86890699525019101656253224030, 0.087060008746063258468802699876, 2.66224526129260916175381458801, 4.03830390245798747729318852242, 5.15148339324423241759456046642, 7.85003655827736469972696227418, 8.302097337345944054118396143889, 9.363857628783182190394654186203, 11.37770643007265864696560078248, 12.58639570806949117343393349073, 13.13342440787641160204612906555

Graph of the ZZ-function along the critical line