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Properties

Label 2-58-29.20-c1-0-1
Degree 22
Conductor 5858
Sign 0.497+0.867i0.497 + 0.867i
Analytic cond. 0.4631320.463132
Root an. cond. 0.6805380.680538
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 + 0.433i)2-s + (−1.52 − 1.90i)3-s + (0.623 − 0.781i)4-s + (2.60 − 1.25i)5-s + (2.19 + 1.05i)6-s + (−1.89 − 2.37i)7-s + (−0.222 + 0.974i)8-s + (−0.658 + 2.88i)9-s + (−1.79 + 2.25i)10-s + (1.22 + 5.34i)11-s − 2.44·12-s + (0.0239 + 0.104i)13-s + (2.74 + 1.32i)14-s + (−6.34 − 3.05i)15-s + (−0.222 − 0.974i)16-s + 0.816·17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (−0.878 − 1.10i)3-s + (0.311 − 0.390i)4-s + (1.16 − 0.559i)5-s + (0.897 + 0.432i)6-s + (−0.717 − 0.899i)7-s + (−0.0786 + 0.344i)8-s + (−0.219 + 0.961i)9-s + (−0.568 + 0.713i)10-s + (0.367 + 1.61i)11-s − 0.704·12-s + (0.00663 + 0.0290i)13-s + (0.732 + 0.352i)14-s + (−1.63 − 0.789i)15-s + (−0.0556 − 0.243i)16-s + 0.197·17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((0.497+0.867i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1/2) L(s)=((0.497+0.867i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: 0.497+0.867i0.497 + 0.867i
Analytic conductor: 0.4631320.463132
Root analytic conductor: 0.6805380.680538
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ58(49,⋅)\chi_{58} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1/2), 0.497+0.867i)(2,\ 58,\ (\ :1/2),\ 0.497 + 0.867i)

Particular Values

L(1)L(1) ≈\approx 0.505998−0.292961i0.505998 - 0.292961i
L(12)L(\frac12) ≈\approx 0.505998−0.292961i0.505998 - 0.292961i
L(32)L(\frac{3}{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+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
29 1+(5.37+0.304i)T 1 + (5.37 + 0.304i)T
good3 1+(1.52+1.90i)T+(−0.667+2.92i)T2 1 + (1.52 + 1.90i)T + (-0.667 + 2.92i)T^{2}
5 1+(−2.60+1.25i)T+(3.11−3.90i)T2 1 + (-2.60 + 1.25i)T + (3.11 - 3.90i)T^{2}
7 1+(1.89+2.37i)T+(−1.55+6.82i)T2 1 + (1.89 + 2.37i)T + (-1.55 + 6.82i)T^{2}
11 1+(−1.22−5.34i)T+(−9.91+4.77i)T2 1 + (-1.22 - 5.34i)T + (-9.91 + 4.77i)T^{2}
13 1+(−0.0239−0.104i)T+(−11.7+5.64i)T2 1 + (-0.0239 - 0.104i)T + (-11.7 + 5.64i)T^{2}
17 1−0.816T+17T2 1 - 0.816T + 17T^{2}
19 1+(−1.27+1.59i)T+(−4.22−18.5i)T2 1 + (-1.27 + 1.59i)T + (-4.22 - 18.5i)T^{2}
23 1+(−8.25−3.97i)T+(14.3+17.9i)T2 1 + (-8.25 - 3.97i)T + (14.3 + 17.9i)T^{2}
31 1+(−3.10+1.49i)T+(19.3−24.2i)T2 1 + (-3.10 + 1.49i)T + (19.3 - 24.2i)T^{2}
37 1+(1.31−5.75i)T+(−33.3−16.0i)T2 1 + (1.31 - 5.75i)T + (-33.3 - 16.0i)T^{2}
41 1+2.43T+41T2 1 + 2.43T + 41T^{2}
43 1+(4.94+2.37i)T+(26.8+33.6i)T2 1 + (4.94 + 2.37i)T + (26.8 + 33.6i)T^{2}
47 1+(1.31+5.76i)T+(−42.3+20.3i)T2 1 + (1.31 + 5.76i)T + (-42.3 + 20.3i)T^{2}
53 1+(3.55−1.71i)T+(33.0−41.4i)T2 1 + (3.55 - 1.71i)T + (33.0 - 41.4i)T^{2}
59 1−1.13T+59T2 1 - 1.13T + 59T^{2}
61 1+(−5.09−6.38i)T+(−13.5+59.4i)T2 1 + (-5.09 - 6.38i)T + (-13.5 + 59.4i)T^{2}
67 1+(0.212−0.932i)T+(−60.3−29.0i)T2 1 + (0.212 - 0.932i)T + (-60.3 - 29.0i)T^{2}
71 1+(0.531+2.32i)T+(−63.9+30.8i)T2 1 + (0.531 + 2.32i)T + (-63.9 + 30.8i)T^{2}
73 1+(8.01+3.85i)T+(45.5+57.0i)T2 1 + (8.01 + 3.85i)T + (45.5 + 57.0i)T^{2}
79 1+(−0.934+4.09i)T+(−71.1−34.2i)T2 1 + (-0.934 + 4.09i)T + (-71.1 - 34.2i)T^{2}
83 1+(9.64−12.0i)T+(−18.4−80.9i)T2 1 + (9.64 - 12.0i)T + (-18.4 - 80.9i)T^{2}
89 1+(4.99−2.40i)T+(55.4−69.5i)T2 1 + (4.99 - 2.40i)T + (55.4 - 69.5i)T^{2}
97 1+(−7.43+9.32i)T+(−21.5−94.5i)T2 1 + (-7.43 + 9.32i)T + (-21.5 - 94.5i)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

−15.13088816902901066764033821741, −13.47313295606733456371548808576, −12.94330913951854852471480860346, −11.68122478336663530211977701313, −10.14183016051719082352994796072, −9.324038968592998818674907696742, −7.28708893006793575825702581359, −6.66206929019848200283755473434, −5.26531205183255138421576235460, −1.46253132642499295750824365909, 3.11927520482895634263698173579, 5.54087034181358025019017811898, 6.39266598552677948046117087052, 8.861291556988982830764991044098, 9.714271284552490859378242950739, 10.68450130722518723639181048828, 11.49272491730028602910923243900, 12.98611579061472976487609281449, 14.43362710956709640046237494933, 15.76864140591131721085978036823

Graph of the ZZ-function along the critical line