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Properties

Label 2-58-29.20-c1-0-0
Degree 22
Conductor 5858
Sign 0.614−0.788i0.614 - 0.788i
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 + (1.02 + 1.28i)3-s + (0.623 − 0.781i)4-s + (−1.07 + 0.517i)5-s + (−1.47 − 0.711i)6-s + (1.27 + 1.59i)7-s + (−0.222 + 0.974i)8-s + (0.0699 − 0.306i)9-s + (0.744 − 0.933i)10-s + (−0.819 − 3.59i)11-s + 1.63·12-s + (−0.479 − 2.10i)13-s + (−1.84 − 0.886i)14-s + (−1.76 − 0.848i)15-s + (−0.222 − 0.974i)16-s − 6.53·17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.589 + 0.739i)3-s + (0.311 − 0.390i)4-s + (−0.481 + 0.231i)5-s + (−0.602 − 0.290i)6-s + (0.481 + 0.603i)7-s + (−0.0786 + 0.344i)8-s + (0.0233 − 0.102i)9-s + (0.235 − 0.295i)10-s + (−0.247 − 1.08i)11-s + 0.473·12-s + (−0.133 − 0.583i)13-s + (−0.492 − 0.236i)14-s + (−0.455 − 0.219i)15-s + (−0.0556 − 0.243i)16-s − 1.58·17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((0.614−0.788i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1/2) L(s)=((0.614−0.788i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: 0.614−0.788i0.614 - 0.788i
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), 0.614−0.788i)(2,\ 58,\ (\ :1/2),\ 0.614 - 0.788i)

Particular Values

L(1)L(1) ≈\approx 0.672759+0.328599i0.672759 + 0.328599i
L(12)L(\frac12) ≈\approx 0.672759+0.328599i0.672759 + 0.328599i
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+(−1.75−5.09i)T 1 + (-1.75 - 5.09i)T
good3 1+(−1.02−1.28i)T+(−0.667+2.92i)T2 1 + (-1.02 - 1.28i)T + (-0.667 + 2.92i)T^{2}
5 1+(1.07−0.517i)T+(3.11−3.90i)T2 1 + (1.07 - 0.517i)T + (3.11 - 3.90i)T^{2}
7 1+(−1.27−1.59i)T+(−1.55+6.82i)T2 1 + (-1.27 - 1.59i)T + (-1.55 + 6.82i)T^{2}
11 1+(0.819+3.59i)T+(−9.91+4.77i)T2 1 + (0.819 + 3.59i)T + (-9.91 + 4.77i)T^{2}
13 1+(0.479+2.10i)T+(−11.7+5.64i)T2 1 + (0.479 + 2.10i)T + (-11.7 + 5.64i)T^{2}
17 1+6.53T+17T2 1 + 6.53T + 17T^{2}
19 1+(−3.31+4.15i)T+(−4.22−18.5i)T2 1 + (-3.31 + 4.15i)T + (-4.22 - 18.5i)T^{2}
23 1+(−5.30−2.55i)T+(14.3+17.9i)T2 1 + (-5.30 - 2.55i)T + (14.3 + 17.9i)T^{2}
31 1+(8.10−3.90i)T+(19.3−24.2i)T2 1 + (8.10 - 3.90i)T + (19.3 - 24.2i)T^{2}
37 1+(0.406−1.78i)T+(−33.3−16.0i)T2 1 + (0.406 - 1.78i)T + (-33.3 - 16.0i)T^{2}
41 1+8.32T+41T2 1 + 8.32T + 41T^{2}
43 1+(−3.31−1.59i)T+(26.8+33.6i)T2 1 + (-3.31 - 1.59i)T + (26.8 + 33.6i)T^{2}
47 1+(−0.220−0.967i)T+(−42.3+20.3i)T2 1 + (-0.220 - 0.967i)T + (-42.3 + 20.3i)T^{2}
53 1+(−5.10+2.45i)T+(33.0−41.4i)T2 1 + (-5.10 + 2.45i)T + (33.0 - 41.4i)T^{2}
59 1−2.94T+59T2 1 - 2.94T + 59T^{2}
61 1+(1.12+1.41i)T+(−13.5+59.4i)T2 1 + (1.12 + 1.41i)T + (-13.5 + 59.4i)T^{2}
67 1+(1.34−5.89i)T+(−60.3−29.0i)T2 1 + (1.34 - 5.89i)T + (-60.3 - 29.0i)T^{2}
71 1+(0.836+3.66i)T+(−63.9+30.8i)T2 1 + (0.836 + 3.66i)T + (-63.9 + 30.8i)T^{2}
73 1+(11.4+5.52i)T+(45.5+57.0i)T2 1 + (11.4 + 5.52i)T + (45.5 + 57.0i)T^{2}
79 1+(3.14−13.7i)T+(−71.1−34.2i)T2 1 + (3.14 - 13.7i)T + (-71.1 - 34.2i)T^{2}
83 1+(−1.27+1.60i)T+(−18.4−80.9i)T2 1 + (-1.27 + 1.60i)T + (-18.4 - 80.9i)T^{2}
89 1+(−14.8+7.16i)T+(55.4−69.5i)T2 1 + (-14.8 + 7.16i)T + (55.4 - 69.5i)T^{2}
97 1+(2.86−3.59i)T+(−21.5−94.5i)T2 1 + (2.86 - 3.59i)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

−15.42405112315332765079705485417, −14.74709454999103112137187076041, −13.34236243955626325862185173298, −11.54532389285365864472748130329, −10.71084515642885466165705077048, −9.147299650041159251404268489018, −8.576169814825708842112054332374, −7.06390817293857783222318047311, −5.20735773581590030516994544317, −3.17937074556836729539283677817, 2.04754554209974826976360601438, 4.39231051012775475224377202971, 7.05503994162231421680785614924, 7.79567785121068913314616494151, 8.963964115106636420596499608253, 10.40627171126045814272768096541, 11.63000226753219377083002232428, 12.76695450116640065944895007194, 13.76181712033674516475855588533, 15.01687231861016030508341883882

Graph of the ZZ-function along the critical line