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Properties

Label 2-58-29.16-c1-0-1
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(45,⋅)\chi_{58} (45, \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

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

Graph of the ZZ-function along the critical line