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Properties

Label 2-59-59.10-c6-0-19
Degree 22
Conductor 5959
Sign −0.0482+0.998i-0.0482 + 0.998i
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  + (−14.5 − 5.78i)2-s + (45.9 + 5.00i)3-s + (130. + 123. i)4-s + (99.9 − 117. i)5-s + (−638. − 338. i)6-s + (−238. − 143. i)7-s + (−760. − 1.64e3i)8-s + (1.37e3 + 303. i)9-s + (−2.13e3 + 1.12e3i)10-s + (1.15e3 + 62.7i)11-s + (5.38e3 + 6.34e3i)12-s + (−651. − 2.96e3i)13-s + (2.63e3 + 3.46e3i)14-s + (5.18e3 − 4.91e3i)15-s + (907. + 1.67e4i)16-s + (−3.39e3 + 2.04e3i)17-s + ⋯
L(s)  = 1  + (−1.81 − 0.722i)2-s + (1.70 + 0.185i)3-s + (2.04 + 1.93i)4-s + (0.799 − 0.941i)5-s + (−2.95 − 1.56i)6-s + (−0.696 − 0.419i)7-s + (−1.48 − 3.20i)8-s + (1.88 + 0.415i)9-s + (−2.13 + 1.12i)10-s + (0.869 + 0.0471i)11-s + (3.11 + 3.67i)12-s + (−0.296 − 1.34i)13-s + (0.960 + 1.26i)14-s + (1.53 − 1.45i)15-s + (0.221 + 4.08i)16-s + (−0.691 + 0.416i)17-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=((−0.0482+0.998i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=((−0.0482+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: −0.0482+0.998i-0.0482 + 0.998i
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), −0.0482+0.998i)(2,\ 59,\ (\ :3),\ -0.0482 + 0.998i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.05126−1.10324i1.05126 - 1.10324i
L(12)L(\frac12) ≈\approx 1.05126−1.10324i1.05126 - 1.10324i
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+(−1.98e5+5.25e4i)T 1 + (-1.98e5 + 5.25e4i)T
good2 1+(14.5+5.78i)T+(46.4+44.0i)T2 1 + (14.5 + 5.78i)T + (46.4 + 44.0i)T^{2}
3 1+(−45.9−5.00i)T+(711.+156.i)T2 1 + (-45.9 - 5.00i)T + (711. + 156. i)T^{2}
5 1+(−99.9+117.i)T+(−2.52e3−1.54e4i)T2 1 + (-99.9 + 117. i)T + (-2.52e3 - 1.54e4i)T^{2}
7 1+(238.+143.i)T+(5.51e4+1.03e5i)T2 1 + (238. + 143. i)T + (5.51e4 + 1.03e5i)T^{2}
11 1+(−1.15e3−62.7i)T+(1.76e6+1.91e5i)T2 1 + (-1.15e3 - 62.7i)T + (1.76e6 + 1.91e5i)T^{2}
13 1+(651.+2.96e3i)T+(−4.38e6+2.02e6i)T2 1 + (651. + 2.96e3i)T + (-4.38e6 + 2.02e6i)T^{2}
17 1+(3.39e3−2.04e3i)T+(1.13e7−2.13e7i)T2 1 + (3.39e3 - 2.04e3i)T + (1.13e7 - 2.13e7i)T^{2}
19 1+(9.18−33.0i)T+(−4.03e7−2.42e7i)T2 1 + (9.18 - 33.0i)T + (-4.03e7 - 2.42e7i)T^{2}
23 1+(−1.12e4+7.66e3i)T+(5.47e7−1.37e8i)T2 1 + (-1.12e4 + 7.66e3i)T + (5.47e7 - 1.37e8i)T^{2}
29 1+(−5.99e3−1.50e4i)T+(−4.31e8+4.09e8i)T2 1 + (-5.99e3 - 1.50e4i)T + (-4.31e8 + 4.09e8i)T^{2}
31 1+(1.20e4−3.34e3i)T+(7.60e8−4.57e8i)T2 1 + (1.20e4 - 3.34e3i)T + (7.60e8 - 4.57e8i)T^{2}
37 1+(−2.98e4+6.44e4i)T+(−1.66e9−1.95e9i)T2 1 + (-2.98e4 + 6.44e4i)T + (-1.66e9 - 1.95e9i)T^{2}
41 1+(−1.17e4+1.73e4i)T+(−1.75e9−4.41e9i)T2 1 + (-1.17e4 + 1.73e4i)T + (-1.75e9 - 4.41e9i)T^{2}
43 1+(8.76e4−4.75e3i)T+(6.28e9−6.83e8i)T2 1 + (8.76e4 - 4.75e3i)T + (6.28e9 - 6.83e8i)T^{2}
47 1+(−2.27e4+1.93e4i)T+(1.74e9−1.06e10i)T2 1 + (-2.27e4 + 1.93e4i)T + (1.74e9 - 1.06e10i)T^{2}
53 1+(6.20e4−1.17e5i)T+(−1.24e10−1.83e10i)T2 1 + (6.20e4 - 1.17e5i)T + (-1.24e10 - 1.83e10i)T^{2}
61 1+(−3.35e5−1.33e5i)T+(3.74e10+3.54e10i)T2 1 + (-3.35e5 - 1.33e5i)T + (3.74e10 + 3.54e10i)T^{2}
67 1+(8.86e4+1.91e5i)T+(−5.85e10+6.89e10i)T2 1 + (8.86e4 + 1.91e5i)T + (-5.85e10 + 6.89e10i)T^{2}
71 1+(−1.32e5−1.55e5i)T+(−2.07e10+1.26e11i)T2 1 + (-1.32e5 - 1.55e5i)T + (-2.07e10 + 1.26e11i)T^{2}
73 1+(−2.97e4−3.91e4i)T+(−4.04e10+1.45e11i)T2 1 + (-2.97e4 - 3.91e4i)T + (-4.04e10 + 1.45e11i)T^{2}
79 1+(−5.09e5+5.53e4i)T+(2.37e11−5.22e10i)T2 1 + (-5.09e5 + 5.53e4i)T + (2.37e11 - 5.22e10i)T^{2}
83 1+(−4.13e4−1.22e5i)T+(−2.60e11+1.97e11i)T2 1 + (-4.13e4 - 1.22e5i)T + (-2.60e11 + 1.97e11i)T^{2}
89 1+(1.12e5−4.48e4i)T+(3.60e11−3.41e11i)T2 1 + (1.12e5 - 4.48e4i)T + (3.60e11 - 3.41e11i)T^{2}
97 1+(1.01e6−1.33e6i)T+(−2.22e11−8.02e11i)T2 1 + (1.01e6 - 1.33e6i)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

−13.17842045126496153995780461923, −12.63422407433817777675024792239, −10.60436104101054736292192896035, −9.651907979434952911967441456429, −9.037705862847888199579775450092, −8.268880168201946092844095818086, −6.96567573540581604665915134954, −3.56328530672497930613245054611, −2.30505372631906119674211350012, −0.972033458043886487736252897757, 1.75296324935313023634711915907, 2.74847386742283004333190029810, 6.51606669121723199816699419985, 7.01984644826513738921377128293, 8.492510089821092042047482959177, 9.480728369614491140511067608330, 9.714438142429367538915748396396, 11.40319686717266526557860917335, 13.66899702158180470534919850970, 14.57191681257100252040687120642

Graph of the ZZ-function along the critical line