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Properties

Label 2-58-29.11-c2-0-4
Degree 22
Conductor 5858
Sign 0.216+0.976i0.216 + 0.976i
Analytic cond. 1.580381.58038
Root an. cond. 1.257131.25713
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 − 0.467i)2-s + (−0.558 − 4.96i)3-s + (1.56 − 1.24i)4-s + (−2.91 + 6.04i)5-s + (−3.06 − 6.36i)6-s + (4.81 − 6.03i)7-s + (1.50 − 2.39i)8-s + (−15.5 + 3.54i)9-s + (−1.06 + 9.43i)10-s + (5.19 + 8.26i)11-s + (−7.05 − 7.05i)12-s + (19.8 + 4.52i)13-s + (3.60 − 10.3i)14-s + (31.6 + 11.0i)15-s + (0.890 − 3.89i)16-s + (−13.5 + 13.5i)17-s + ⋯
L(s)  = 1  + (0.667 − 0.233i)2-s + (−0.186 − 1.65i)3-s + (0.390 − 0.311i)4-s + (−0.582 + 1.20i)5-s + (−0.510 − 1.06i)6-s + (0.687 − 0.862i)7-s + (0.188 − 0.299i)8-s + (−1.72 + 0.393i)9-s + (−0.106 + 0.943i)10-s + (0.472 + 0.751i)11-s + (−0.588 − 0.588i)12-s + (1.52 + 0.347i)13-s + (0.257 − 0.735i)14-s + (2.10 + 0.737i)15-s + (0.0556 − 0.243i)16-s + (−0.797 + 0.797i)17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((0.216+0.976i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1) L(s)=((0.216+0.976i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: 0.216+0.976i0.216 + 0.976i
Analytic conductor: 1.580381.58038
Root analytic conductor: 1.257131.25713
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ58(11,⋅)\chi_{58} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1), 0.216+0.976i)(2,\ 58,\ (\ :1),\ 0.216 + 0.976i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.19153−0.955921i1.19153 - 0.955921i
L(12)L(\frac12) ≈\approx 1.19153−0.955921i1.19153 - 0.955921i
L(2)L(2) 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)
bad2 1+(−1.33+0.467i)T 1 + (-1.33 + 0.467i)T
29 1+(−8.87+27.6i)T 1 + (-8.87 + 27.6i)T
good3 1+(0.558+4.96i)T+(−8.77+2.00i)T2 1 + (0.558 + 4.96i)T + (-8.77 + 2.00i)T^{2}
5 1+(2.91−6.04i)T+(−15.5−19.5i)T2 1 + (2.91 - 6.04i)T + (-15.5 - 19.5i)T^{2}
7 1+(−4.81+6.03i)T+(−10.9−47.7i)T2 1 + (-4.81 + 6.03i)T + (-10.9 - 47.7i)T^{2}
11 1+(−5.19−8.26i)T+(−52.4+109.i)T2 1 + (-5.19 - 8.26i)T + (-52.4 + 109. i)T^{2}
13 1+(−19.8−4.52i)T+(152.+73.3i)T2 1 + (-19.8 - 4.52i)T + (152. + 73.3i)T^{2}
17 1+(13.5−13.5i)T−289iT2 1 + (13.5 - 13.5i)T - 289iT^{2}
19 1+(16.3+1.84i)T+(351.+80.3i)T2 1 + (16.3 + 1.84i)T + (351. + 80.3i)T^{2}
23 1+(2.47−1.19i)T+(329.−413.i)T2 1 + (2.47 - 1.19i)T + (329. - 413. i)T^{2}
31 1+(39.3−13.7i)T+(751.−599.i)T2 1 + (39.3 - 13.7i)T + (751. - 599. i)T^{2}
37 1+(17.4−27.7i)T+(−593.−1.23e3i)T2 1 + (17.4 - 27.7i)T + (-593. - 1.23e3i)T^{2}
41 1+(−5.70−5.70i)T+1.68e3iT2 1 + (-5.70 - 5.70i)T + 1.68e3iT^{2}
43 1+(−18.8+53.9i)T+(−1.44e3−1.15e3i)T2 1 + (-18.8 + 53.9i)T + (-1.44e3 - 1.15e3i)T^{2}
47 1+(37.3−23.4i)T+(958.−1.99e3i)T2 1 + (37.3 - 23.4i)T + (958. - 1.99e3i)T^{2}
53 1+(8.97+4.32i)T+(1.75e3+2.19e3i)T2 1 + (8.97 + 4.32i)T + (1.75e3 + 2.19e3i)T^{2}
59 1−21.7T+3.48e3T2 1 - 21.7T + 3.48e3T^{2}
61 1+(9.69+86.0i)T+(−3.62e3+828.i)T2 1 + (9.69 + 86.0i)T + (-3.62e3 + 828. i)T^{2}
67 1+(−66.2+15.1i)T+(4.04e3−1.94e3i)T2 1 + (-66.2 + 15.1i)T + (4.04e3 - 1.94e3i)T^{2}
71 1+(−26.7−6.09i)T+(4.54e3+2.18e3i)T2 1 + (-26.7 - 6.09i)T + (4.54e3 + 2.18e3i)T^{2}
73 1+(23.2+8.13i)T+(4.16e3+3.32e3i)T2 1 + (23.2 + 8.13i)T + (4.16e3 + 3.32e3i)T^{2}
79 1+(−44.4−27.9i)T+(2.70e3+5.62e3i)T2 1 + (-44.4 - 27.9i)T + (2.70e3 + 5.62e3i)T^{2}
83 1+(−33.1−41.5i)T+(−1.53e3+6.71e3i)T2 1 + (-33.1 - 41.5i)T + (-1.53e3 + 6.71e3i)T^{2}
89 1+(127.−44.5i)T+(6.19e3−4.93e3i)T2 1 + (127. - 44.5i)T + (6.19e3 - 4.93e3i)T^{2}
97 1+(19.4−172.i)T+(−9.17e3−2.09e3i)T2 1 + (19.4 - 172. i)T + (-9.17e3 - 2.09e3i)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

−14.32726251751923327846175614184, −13.58030952530955789804366855565, −12.53538089207946015755867535358, −11.30650862871011182706674619520, −10.83622010327270371317277447294, −8.180600966214167945575756939091, −6.99529360908554076886630524601, −6.39331270611461636047789913195, −3.93549581021981710437633976504, −1.80765256299452518221139545817, 3.75338256048516801255937575543, 4.80788079235937435581206078429, 5.80967559150555820972094689401, 8.536313792516358365300646386945, 8.941352615524986889283087169873, 10.95075285897414143163323344743, 11.56666304562266115837808621879, 12.91786845752363175006279488626, 14.38961343368066209026714258351, 15.40312121164099131675728171690

Graph of the ZZ-function along the critical line