↑

Properties

Label 2-59-59.11-c6-0-16
Degree 22
Conductor 5959
Sign −0.980−0.196i-0.980 - 0.196i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.82 − 12.8i)2-s + (−6.70 + 40.9i)3-s + (−98.2 + 45.4i)4-s + (−36.2 + 130. i)5-s + (543. − 29.4i)6-s + (400. − 379. i)7-s + (351. + 462. i)8-s + (−938. − 316. i)9-s + (1.77e3 + 96.3i)10-s + (−1.42e3 + 1.20e3i)11-s + (−1.20e3 − 4.32e3i)12-s + (15.4 + 45.9i)13-s + (−5.99e3 − 4.06e3i)14-s + (−5.10e3 − 2.35e3i)15-s + (454. − 534. i)16-s + (−404. − 382. i)17-s + ⋯
L(s)  = 1  + (−0.352 − 1.60i)2-s + (−0.248 + 1.51i)3-s + (−1.53 + 0.710i)4-s + (−0.290 + 1.04i)5-s + (2.51 − 0.136i)6-s + (1.16 − 1.10i)7-s + (0.687 + 0.904i)8-s + (−1.28 − 0.433i)9-s + (1.77 + 0.0963i)10-s + (−1.06 + 0.906i)11-s + (−0.695 − 2.50i)12-s + (0.00704 + 0.0209i)13-s + (−2.18 − 1.48i)14-s + (−1.51 − 0.699i)15-s + (0.110 − 0.130i)16-s + (−0.0822 − 0.0779i)17-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=((−0.980−0.196i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=((−0.980−0.196i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: −0.980−0.196i-0.980 - 0.196i
Analytic conductor: 13.573113.5731
Root analytic conductor: 3.684183.68418
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ59(11,⋅)\chi_{59} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 59, ( :3), −0.980−0.196i)(2,\ 59,\ (\ :3),\ -0.980 - 0.196i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.0122277+0.123044i0.0122277 + 0.123044i
L(12)L(\frac12) ≈\approx 0.0122277+0.123044i0.0122277 + 0.123044i
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+(1.28e5−1.60e5i)T 1 + (1.28e5 - 1.60e5i)T
good2 1+(2.82+12.8i)T+(−58.0+26.8i)T2 1 + (2.82 + 12.8i)T + (-58.0 + 26.8i)T^{2}
3 1+(6.70−40.9i)T+(−690.−232.i)T2 1 + (6.70 - 40.9i)T + (-690. - 232. i)T^{2}
5 1+(36.2−130.i)T+(−1.33e4−8.05e3i)T2 1 + (36.2 - 130. i)T + (-1.33e4 - 8.05e3i)T^{2}
7 1+(−400.+379.i)T+(6.36e3−1.17e5i)T2 1 + (-400. + 379. i)T + (6.36e3 - 1.17e5i)T^{2}
11 1+(1.42e3−1.20e3i)T+(2.86e5−1.74e6i)T2 1 + (1.42e3 - 1.20e3i)T + (2.86e5 - 1.74e6i)T^{2}
13 1+(−15.4−45.9i)T+(−3.84e6+2.92e6i)T2 1 + (-15.4 - 45.9i)T + (-3.84e6 + 2.92e6i)T^{2}
17 1+(404.+382.i)T+(1.30e6+2.41e7i)T2 1 + (404. + 382. i)T + (1.30e6 + 2.41e7i)T^{2}
19 1+(3.05e3+7.67e3i)T+(−3.41e7+3.23e7i)T2 1 + (3.05e3 + 7.67e3i)T + (-3.41e7 + 3.23e7i)T^{2}
23 1+(2.12e3+1.95e4i)T+(−1.44e8+3.18e7i)T2 1 + (2.12e3 + 1.95e4i)T + (-1.44e8 + 3.18e7i)T^{2}
29 1+(2.56e4+5.63e3i)T+(5.39e8+2.49e8i)T2 1 + (2.56e4 + 5.63e3i)T + (5.39e8 + 2.49e8i)T^{2}
31 1+(5.17e4+2.06e4i)T+(6.44e8+6.10e8i)T2 1 + (5.17e4 + 2.06e4i)T + (6.44e8 + 6.10e8i)T^{2}
37 1+(−3.08e4+4.05e4i)T+(−6.86e8−2.47e9i)T2 1 + (-3.08e4 + 4.05e4i)T + (-6.86e8 - 2.47e9i)T^{2}
41 1+(5.96e4+6.48e3i)T+(4.63e9+1.02e9i)T2 1 + (5.96e4 + 6.48e3i)T + (4.63e9 + 1.02e9i)T^{2}
43 1+(−5.61e4−4.77e4i)T+(1.02e9+6.23e9i)T2 1 + (-5.61e4 - 4.77e4i)T + (1.02e9 + 6.23e9i)T^{2}
47 1+(−5.18e4+1.43e4i)T+(9.23e9−5.55e9i)T2 1 + (-5.18e4 + 1.43e4i)T + (9.23e9 - 5.55e9i)T^{2}
53 1+(−3.03e3−5.60e4i)T+(−2.20e10+2.39e9i)T2 1 + (-3.03e3 - 5.60e4i)T + (-2.20e10 + 2.39e9i)T^{2}
61 1+(−5.19e3−2.35e4i)T+(−4.67e10+2.16e10i)T2 1 + (-5.19e3 - 2.35e4i)T + (-4.67e10 + 2.16e10i)T^{2}
67 1+(1.01e5+1.33e5i)T+(−2.42e10+8.71e10i)T2 1 + (1.01e5 + 1.33e5i)T + (-2.42e10 + 8.71e10i)T^{2}
71 1+(−1.89e4−6.82e4i)T+(−1.09e11+6.60e10i)T2 1 + (-1.89e4 - 6.82e4i)T + (-1.09e11 + 6.60e10i)T^{2}
73 1+(4.13e5+2.80e5i)T+(5.60e10+1.40e11i)T2 1 + (4.13e5 + 2.80e5i)T + (5.60e10 + 1.40e11i)T^{2}
79 1+(−2.67e4−1.63e5i)T+(−2.30e11+7.76e10i)T2 1 + (-2.67e4 - 1.63e5i)T + (-2.30e11 + 7.76e10i)T^{2}
83 1+(−1.38e5+7.34e4i)T+(1.83e11−2.70e11i)T2 1 + (-1.38e5 + 7.34e4i)T + (1.83e11 - 2.70e11i)T^{2}
89 1+(1.69e5−7.71e5i)T+(−4.51e11−2.08e11i)T2 1 + (1.69e5 - 7.71e5i)T + (-4.51e11 - 2.08e11i)T^{2}
97 1+(−1.42e6+9.64e5i)T+(3.08e11−7.73e11i)T2 1 + (-1.42e6 + 9.64e5i)T + (3.08e11 - 7.73e11i)T^{2}
show more
show less
   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

−13.02580396903947937898685662530, −11.39092648722880016383302449484, −10.64004095242776741133306282128, −10.53876617814802113466365221669, −9.183917601021547249236841687909, −7.49566503106718369491653363419, −4.73513090366247056201588688150, −3.93646941572441041755944167825, −2.41237913626229515419307968075, −0.05942118299399765262501772696, 1.61356285318745362349027305729, 5.30815218210106867742381652244, 5.80007540630520269729354747164, 7.52535299467922964068514748399, 8.160063440744229902876666486752, 8.868143233894852855715879498554, 11.41726538744578820507539586282, 12.50306835313968830044559452830, 13.46767620461895159982222944577, 14.61230325981254222590118665113

Graph of the ZZ-function along the critical line