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Properties

Label 2-59-59.11-c6-0-23
Degree 22
Conductor 5959
Sign −0.329+0.944i-0.329 + 0.944i
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  + (2.64 + 11.9i)2-s + (7.84 − 47.8i)3-s + (−78.8 + 36.4i)4-s + (37.8 − 136. i)5-s + (595. − 32.2i)6-s + (−396. + 375. i)7-s + (−170. − 223. i)8-s + (−1.53e3 − 518. i)9-s + (1.73e3 + 94.0i)10-s + (−129. + 109. i)11-s + (1.12e3 + 4.06e3i)12-s + (−952. − 2.82e3i)13-s + (−5.55e3 − 3.76e3i)14-s + (−6.22e3 − 2.88e3i)15-s + (−1.36e3 + 1.60e3i)16-s + (−6.30e3 − 5.97e3i)17-s + ⋯
L(s)  = 1  + (0.330 + 1.49i)2-s + (0.290 − 1.77i)3-s + (−1.23 + 0.569i)4-s + (0.302 − 1.09i)5-s + (2.75 − 0.149i)6-s + (−1.15 + 1.09i)7-s + (−0.332 − 0.436i)8-s + (−2.11 − 0.711i)9-s + (1.73 + 0.0940i)10-s + (−0.0971 + 0.0825i)11-s + (0.652 + 2.35i)12-s + (−0.433 − 1.28i)13-s + (−2.02 − 1.37i)14-s + (−1.84 − 0.853i)15-s + (−0.333 + 0.392i)16-s + (−1.28 − 1.21i)17-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=((−0.329+0.944i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+3) L(s)=((−0.329+0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: −0.329+0.944i-0.329 + 0.944i
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), −0.329+0.944i)(2,\ 59,\ (\ :3),\ -0.329 + 0.944i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 0.581494−0.819162i0.581494 - 0.819162i
L(12)L(\frac12) ≈\approx 0.581494−0.819162i0.581494 - 0.819162i
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+(−1.41e5−1.49e5i)T 1 + (-1.41e5 - 1.49e5i)T
good2 1+(−2.64−11.9i)T+(−58.0+26.8i)T2 1 + (-2.64 - 11.9i)T + (-58.0 + 26.8i)T^{2}
3 1+(−7.84+47.8i)T+(−690.−232.i)T2 1 + (-7.84 + 47.8i)T + (-690. - 232. i)T^{2}
5 1+(−37.8+136.i)T+(−1.33e4−8.05e3i)T2 1 + (-37.8 + 136. i)T + (-1.33e4 - 8.05e3i)T^{2}
7 1+(396.−375.i)T+(6.36e3−1.17e5i)T2 1 + (396. - 375. i)T + (6.36e3 - 1.17e5i)T^{2}
11 1+(129.−109.i)T+(2.86e5−1.74e6i)T2 1 + (129. - 109. i)T + (2.86e5 - 1.74e6i)T^{2}
13 1+(952.+2.82e3i)T+(−3.84e6+2.92e6i)T2 1 + (952. + 2.82e3i)T + (-3.84e6 + 2.92e6i)T^{2}
17 1+(6.30e3+5.97e3i)T+(1.30e6+2.41e7i)T2 1 + (6.30e3 + 5.97e3i)T + (1.30e6 + 2.41e7i)T^{2}
19 1+(−2.95e3−7.41e3i)T+(−3.41e7+3.23e7i)T2 1 + (-2.95e3 - 7.41e3i)T + (-3.41e7 + 3.23e7i)T^{2}
23 1+(1.02e3+9.41e3i)T+(−1.44e8+3.18e7i)T2 1 + (1.02e3 + 9.41e3i)T + (-1.44e8 + 3.18e7i)T^{2}
29 1+(1.82e4+4.01e3i)T+(5.39e8+2.49e8i)T2 1 + (1.82e4 + 4.01e3i)T + (5.39e8 + 2.49e8i)T^{2}
31 1+(−1.65e4−6.57e3i)T+(6.44e8+6.10e8i)T2 1 + (-1.65e4 - 6.57e3i)T + (6.44e8 + 6.10e8i)T^{2}
37 1+(−4.36e4+5.74e4i)T+(−6.86e8−2.47e9i)T2 1 + (-4.36e4 + 5.74e4i)T + (-6.86e8 - 2.47e9i)T^{2}
41 1+(−3.29e4−3.58e3i)T+(4.63e9+1.02e9i)T2 1 + (-3.29e4 - 3.58e3i)T + (4.63e9 + 1.02e9i)T^{2}
43 1+(3.32e4+2.82e4i)T+(1.02e9+6.23e9i)T2 1 + (3.32e4 + 2.82e4i)T + (1.02e9 + 6.23e9i)T^{2}
47 1+(−3.03e4+8.43e3i)T+(9.23e9−5.55e9i)T2 1 + (-3.03e4 + 8.43e3i)T + (9.23e9 - 5.55e9i)T^{2}
53 1+(7.12e3+1.31e5i)T+(−2.20e10+2.39e9i)T2 1 + (7.12e3 + 1.31e5i)T + (-2.20e10 + 2.39e9i)T^{2}
61 1+(−5.34e4−2.42e5i)T+(−4.67e10+2.16e10i)T2 1 + (-5.34e4 - 2.42e5i)T + (-4.67e10 + 2.16e10i)T^{2}
67 1+(6.60e4+8.69e4i)T+(−2.42e10+8.71e10i)T2 1 + (6.60e4 + 8.69e4i)T + (-2.42e10 + 8.71e10i)T^{2}
71 1+(8.73e4+3.14e5i)T+(−1.09e11+6.60e10i)T2 1 + (8.73e4 + 3.14e5i)T + (-1.09e11 + 6.60e10i)T^{2}
73 1+(−1.96e5−1.33e5i)T+(5.60e10+1.40e11i)T2 1 + (-1.96e5 - 1.33e5i)T + (5.60e10 + 1.40e11i)T^{2}
79 1+(−9.19e3−5.60e4i)T+(−2.30e11+7.76e10i)T2 1 + (-9.19e3 - 5.60e4i)T + (-2.30e11 + 7.76e10i)T^{2}
83 1+(−6.68e4+3.54e4i)T+(1.83e11−2.70e11i)T2 1 + (-6.68e4 + 3.54e4i)T + (1.83e11 - 2.70e11i)T^{2}
89 1+(−2.39e5+1.08e6i)T+(−4.51e11−2.08e11i)T2 1 + (-2.39e5 + 1.08e6i)T + (-4.51e11 - 2.08e11i)T^{2}
97 1+(2.96e5−2.01e5i)T+(3.08e11−7.73e11i)T2 1 + (2.96e5 - 2.01e5i)T + (3.08e11 - 7.73e11i)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

−13.32497415317668768887717170968, −12.94019070866074096578823603173, −12.09068205647013868684601639008, −9.214932181881270186590411779505, −8.361795498198403731606286462926, −7.28817848321328124780837141447, −6.15689575340908345779945122596, −5.37525916016264394760498408127, −2.46269839272148690933244813907, −0.33794619387501389664884068506, 2.60604704522752117570455150089, 3.64227754904575546111338967961, 4.45919732065856466355109146390, 6.68488248643407295116062817673, 9.324577760364064822153673786914, 9.885995974788860478302751084746, 10.77582477227486601669281361365, 11.32258764731085213188692794374, 13.27254580734297111853312915809, 14.00061572283069929013595079798

Graph of the ZZ-function along the critical line