↑

Properties

Label 2-58-29.12-c2-0-3
Degree 22
Conductor 5858
Sign −0.154+0.988i-0.154 + 0.988i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−0.147 − 0.147i)3-s + 2i·4-s − 8.54i·5-s + 0.295i·6-s − 0.295·7-s + (2 − 2i)8-s − 8.95i·9-s + (−8.54 + 8.54i)10-s + (−0.442 − 0.442i)11-s + (0.295 − 0.295i)12-s + 6.54i·13-s + (0.295 + 0.295i)14-s + (−1.26 + 1.26i)15-s − 4·16-s + (8.95 + 8.95i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.0492 − 0.0492i)3-s + 0.5i·4-s − 1.70i·5-s + 0.0492i·6-s − 0.0421·7-s + (0.250 − 0.250i)8-s − 0.995i·9-s + (−0.854 + 0.854i)10-s + (−0.0402 − 0.0402i)11-s + (0.0246 − 0.0246i)12-s + 0.503i·13-s + (0.0210 + 0.0210i)14-s + (−0.0841 + 0.0841i)15-s − 0.250·16-s + (0.526 + 0.526i)17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((−0.154+0.988i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1) L(s)=((−0.154+0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.154 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: −0.154+0.988i-0.154 + 0.988i
Analytic conductor: 1.580381.58038
Root analytic conductor: 1.257131.25713
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ58(41,⋅)\chi_{58} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1), −0.154+0.988i)(2,\ 58,\ (\ :1),\ -0.154 + 0.988i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.576677−0.673526i0.576677 - 0.673526i
L(12)L(\frac12) ≈\approx 0.576677−0.673526i0.576677 - 0.673526i
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+i)T 1 + (1 + i)T
29 1+(26.2−12.3i)T 1 + (26.2 - 12.3i)T
good3 1+(0.147+0.147i)T+9iT2 1 + (0.147 + 0.147i)T + 9iT^{2}
5 1+8.54iT−25T2 1 + 8.54iT - 25T^{2}
7 1+0.295T+49T2 1 + 0.295T + 49T^{2}
11 1+(0.442+0.442i)T+121iT2 1 + (0.442 + 0.442i)T + 121iT^{2}
13 1−6.54iT−169T2 1 - 6.54iT - 169T^{2}
17 1+(−8.95−8.95i)T+289iT2 1 + (-8.95 - 8.95i)T + 289iT^{2}
19 1+(−19.9−19.9i)T+361iT2 1 + (-19.9 - 19.9i)T + 361iT^{2}
23 1−30.1T+529T2 1 - 30.1T + 529T^{2}
31 1+(28.9+28.9i)T+961iT2 1 + (28.9 + 28.9i)T + 961iT^{2}
37 1+(−25.8+25.8i)T−1.36e3iT2 1 + (-25.8 + 25.8i)T - 1.36e3iT^{2}
41 1+(−31.7+31.7i)T−1.68e3iT2 1 + (-31.7 + 31.7i)T - 1.68e3iT^{2}
43 1+(−50.6−50.6i)T+1.84e3iT2 1 + (-50.6 - 50.6i)T + 1.84e3iT^{2}
47 1+(8.98−8.98i)T−2.20e3iT2 1 + (8.98 - 8.98i)T - 2.20e3iT^{2}
53 1−36.0T+2.80e3T2 1 - 36.0T + 2.80e3T^{2}
59 1−17.1T+3.48e3T2 1 - 17.1T + 3.48e3T^{2}
61 1+(0.499+0.499i)T+3.72e3iT2 1 + (0.499 + 0.499i)T + 3.72e3iT^{2}
67 1−113.iT−4.48e3T2 1 - 113. iT - 4.48e3T^{2}
71 1+80.2iT−5.04e3T2 1 + 80.2iT - 5.04e3T^{2}
73 1+(−7.45+7.45i)T−5.32e3iT2 1 + (-7.45 + 7.45i)T - 5.32e3iT^{2}
79 1+(−40.9−40.9i)T+6.24e3iT2 1 + (-40.9 - 40.9i)T + 6.24e3iT^{2}
83 1+104T+6.88e3T2 1 + 104T + 6.88e3T^{2}
89 1+(−43.6−43.6i)T+7.92e3iT2 1 + (-43.6 - 43.6i)T + 7.92e3iT^{2}
97 1+(−55.4+55.4i)T−9.40e3iT2 1 + (-55.4 + 55.4i)T - 9.40e3iT^{2}
show more
show less
   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

−14.61858136936827141306117208714, −12.99201227299513480102769747274, −12.43158289646496061995226188066, −11.36255481706236493767893910875, −9.572621397881206596706801984954, −9.013222949537231786852163600207, −7.66857876481721200958718361485, −5.63180427588050855257209659047, −3.92347475206225640909444950916, −1.15391140507442434004647527706, 2.87526278439503989591921713497, 5.37913960605198300717458070994, 6.96408443919003709028938568203, 7.69101789154214724965280494916, 9.496653190097419979757123431528, 10.66240708688743942675576260855, 11.31886129277607021175986866591, 13.35841387394291212809741046539, 14.36382697296555482385314058420, 15.23582376755031240505541751217

Graph of the ZZ-function along the critical line