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Properties

Label 2-59-59.10-c6-0-14
Degree 22
Conductor 5959
Sign −0.236+0.971i-0.236 + 0.971i
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  + (−11.7 − 4.69i)2-s + (22.7 + 2.47i)3-s + (70.1 + 66.4i)4-s + (−121. + 142. i)5-s + (−256. − 136. i)6-s + (−243. − 146. i)7-s + (−173. − 374. i)8-s + (−198. − 43.7i)9-s + (2.09e3 − 1.11e3i)10-s + (423. + 22.9i)11-s + (1.43e3 + 1.68e3i)12-s + (394. + 1.79e3i)13-s + (2.18e3 + 2.86e3i)14-s + (−3.11e3 + 2.94e3i)15-s + (−51.4 − 948. i)16-s + (4.75e3 − 2.86e3i)17-s + ⋯
L(s)  = 1  + (−1.47 − 0.586i)2-s + (0.844 + 0.0918i)3-s + (1.09 + 1.03i)4-s + (−0.968 + 1.14i)5-s + (−1.18 − 0.629i)6-s + (−0.710 − 0.427i)7-s + (−0.338 − 0.731i)8-s + (−0.272 − 0.0599i)9-s + (2.09 − 1.11i)10-s + (0.318 + 0.0172i)11-s + (0.829 + 0.976i)12-s + (0.179 + 0.814i)13-s + (0.795 + 1.04i)14-s + (−0.922 + 0.873i)15-s + (−0.0125 − 0.231i)16-s + (0.968 − 0.582i)17-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=((−0.236+0.971i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=((−0.236+0.971i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: −0.236+0.971i-0.236 + 0.971i
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), −0.236+0.971i)(2,\ 59,\ (\ :3),\ -0.236 + 0.971i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.319058−0.406230i0.319058 - 0.406230i
L(12)L(\frac12) ≈\approx 0.319058−0.406230i0.319058 - 0.406230i
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+(−2.04e5+1.38e4i)T 1 + (-2.04e5 + 1.38e4i)T
good2 1+(11.7+4.69i)T+(46.4+44.0i)T2 1 + (11.7 + 4.69i)T + (46.4 + 44.0i)T^{2}
3 1+(−22.7−2.47i)T+(711.+156.i)T2 1 + (-22.7 - 2.47i)T + (711. + 156. i)T^{2}
5 1+(121.−142.i)T+(−2.52e3−1.54e4i)T2 1 + (121. - 142. i)T + (-2.52e3 - 1.54e4i)T^{2}
7 1+(243.+146.i)T+(5.51e4+1.03e5i)T2 1 + (243. + 146. i)T + (5.51e4 + 1.03e5i)T^{2}
11 1+(−423.−22.9i)T+(1.76e6+1.91e5i)T2 1 + (-423. - 22.9i)T + (1.76e6 + 1.91e5i)T^{2}
13 1+(−394.−1.79e3i)T+(−4.38e6+2.02e6i)T2 1 + (-394. - 1.79e3i)T + (-4.38e6 + 2.02e6i)T^{2}
17 1+(−4.75e3+2.86e3i)T+(1.13e7−2.13e7i)T2 1 + (-4.75e3 + 2.86e3i)T + (1.13e7 - 2.13e7i)T^{2}
19 1+(−2.83e3+1.02e4i)T+(−4.03e7−2.42e7i)T2 1 + (-2.83e3 + 1.02e4i)T + (-4.03e7 - 2.42e7i)T^{2}
23 1+(−91.8+62.2i)T+(5.47e7−1.37e8i)T2 1 + (-91.8 + 62.2i)T + (5.47e7 - 1.37e8i)T^{2}
29 1+(5.96e3+1.49e4i)T+(−4.31e8+4.09e8i)T2 1 + (5.96e3 + 1.49e4i)T + (-4.31e8 + 4.09e8i)T^{2}
31 1+(−2.29e3+635.i)T+(7.60e8−4.57e8i)T2 1 + (-2.29e3 + 635. i)T + (7.60e8 - 4.57e8i)T^{2}
37 1+(−3.43e4+7.42e4i)T+(−1.66e9−1.95e9i)T2 1 + (-3.43e4 + 7.42e4i)T + (-1.66e9 - 1.95e9i)T^{2}
41 1+(3.63e4−5.35e4i)T+(−1.75e9−4.41e9i)T2 1 + (3.63e4 - 5.35e4i)T + (-1.75e9 - 4.41e9i)T^{2}
43 1+(3.02e4−1.63e3i)T+(6.28e9−6.83e8i)T2 1 + (3.02e4 - 1.63e3i)T + (6.28e9 - 6.83e8i)T^{2}
47 1+(−9.91e4+8.41e4i)T+(1.74e9−1.06e10i)T2 1 + (-9.91e4 + 8.41e4i)T + (1.74e9 - 1.06e10i)T^{2}
53 1+(3.80e3−7.18e3i)T+(−1.24e10−1.83e10i)T2 1 + (3.80e3 - 7.18e3i)T + (-1.24e10 - 1.83e10i)T^{2}
61 1+(1.15e5+4.60e4i)T+(3.74e10+3.54e10i)T2 1 + (1.15e5 + 4.60e4i)T + (3.74e10 + 3.54e10i)T^{2}
67 1+(1.61e5+3.49e5i)T+(−5.85e10+6.89e10i)T2 1 + (1.61e5 + 3.49e5i)T + (-5.85e10 + 6.89e10i)T^{2}
71 1+(−3.56e5−4.19e5i)T+(−2.07e10+1.26e11i)T2 1 + (-3.56e5 - 4.19e5i)T + (-2.07e10 + 1.26e11i)T^{2}
73 1+(2.60e5+3.42e5i)T+(−4.04e10+1.45e11i)T2 1 + (2.60e5 + 3.42e5i)T + (-4.04e10 + 1.45e11i)T^{2}
79 1+(4.43e5−4.82e4i)T+(2.37e11−5.22e10i)T2 1 + (4.43e5 - 4.82e4i)T + (2.37e11 - 5.22e10i)T^{2}
83 1+(−2.64e5−7.83e5i)T+(−2.60e11+1.97e11i)T2 1 + (-2.64e5 - 7.83e5i)T + (-2.60e11 + 1.97e11i)T^{2}
89 1+(−5.09e5+2.02e5i)T+(3.60e11−3.41e11i)T2 1 + (-5.09e5 + 2.02e5i)T + (3.60e11 - 3.41e11i)T^{2}
97 1+(−4.65e5+6.12e5i)T+(−2.22e11−8.02e11i)T2 1 + (-4.65e5 + 6.12e5i)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

−13.73679052405196127803872971449, −11.77642168270055468917105831652, −11.13351158731963766249617291400, −9.851138421725538589157615337335, −9.013236603515977176476739107673, −7.73945329134201013997462143914, −6.91314980393714960391647461275, −3.58908202148077005264250797790, −2.60618712233733359487650274054, −0.36511246355863048265736975772, 1.17551897562177188707886571631, 3.50732123094780966875300034180, 5.82959370988086011602351707067, 7.69190403199463045919323135133, 8.293269639893807611218195653644, 9.080994314976656704367880291553, 10.18581411948739001500739879714, 11.94024996765618355090128701791, 12.97852550932643518932127505563, 14.65768057083491977296527200529

Graph of the ZZ-function along the critical line