↑

Properties

Label 2-58-29.4-c1-0-0
Degree 22
Conductor 5858
Sign 0.863−0.505i0.863 - 0.505i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.626 + 1.30i)3-s + (0.900 + 0.433i)4-s + (−0.308 + 1.35i)5-s + (−0.321 − 1.40i)6-s + (1.78 − 0.861i)7-s + (−0.781 − 0.623i)8-s + (0.568 − 0.712i)9-s + (0.601 − 1.24i)10-s + (−2.70 + 2.15i)11-s + 1.44i·12-s + (−3.81 − 4.77i)13-s + (−1.93 + 0.442i)14-s + (−1.95 + 0.445i)15-s + (0.623 + 0.781i)16-s − 4.25i·17-s + ⋯
L(s)  = 1  + (−0.689 − 0.157i)2-s + (0.361 + 0.751i)3-s + (0.450 + 0.216i)4-s + (−0.137 + 0.604i)5-s + (−0.131 − 0.575i)6-s + (0.676 − 0.325i)7-s + (−0.276 − 0.220i)8-s + (0.189 − 0.237i)9-s + (0.190 − 0.394i)10-s + (−0.814 + 0.649i)11-s + 0.417i·12-s + (−1.05 − 1.32i)13-s + (−0.517 + 0.118i)14-s + (−0.504 + 0.115i)15-s + (0.155 + 0.195i)16-s − 1.03i·17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((0.863−0.505i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1/2) L(s)=((0.863−0.505i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: 0.863−0.505i0.863 - 0.505i
Analytic conductor: 0.4631320.463132
Root analytic conductor: 0.6805380.680538
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ58(33,⋅)\chi_{58} (33, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1/2), 0.863−0.505i)(2,\ 58,\ (\ :1/2),\ 0.863 - 0.505i)

Particular Values

L(1)L(1) ≈\approx 0.699729+0.189730i0.699729 + 0.189730i
L(12)L(\frac12) ≈\approx 0.699729+0.189730i0.699729 + 0.189730i
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+(0.974+0.222i)T 1 + (0.974 + 0.222i)T
29 1+(4.17+3.40i)T 1 + (4.17 + 3.40i)T
good3 1+(−0.626−1.30i)T+(−1.87+2.34i)T2 1 + (-0.626 - 1.30i)T + (-1.87 + 2.34i)T^{2}
5 1+(0.308−1.35i)T+(−4.50−2.16i)T2 1 + (0.308 - 1.35i)T + (-4.50 - 2.16i)T^{2}
7 1+(−1.78+0.861i)T+(4.36−5.47i)T2 1 + (-1.78 + 0.861i)T + (4.36 - 5.47i)T^{2}
11 1+(2.70−2.15i)T+(2.44−10.7i)T2 1 + (2.70 - 2.15i)T + (2.44 - 10.7i)T^{2}
13 1+(3.81+4.77i)T+(−2.89+12.6i)T2 1 + (3.81 + 4.77i)T + (-2.89 + 12.6i)T^{2}
17 1+4.25iT−17T2 1 + 4.25iT - 17T^{2}
19 1+(0.940−1.95i)T+(−11.8−14.8i)T2 1 + (0.940 - 1.95i)T + (-11.8 - 14.8i)T^{2}
23 1+(−0.127−0.556i)T+(−20.7+9.97i)T2 1 + (-0.127 - 0.556i)T + (-20.7 + 9.97i)T^{2}
31 1+(−5.88−1.34i)T+(27.9+13.4i)T2 1 + (-5.88 - 1.34i)T + (27.9 + 13.4i)T^{2}
37 1+(2.12+1.69i)T+(8.23+36.0i)T2 1 + (2.12 + 1.69i)T + (8.23 + 36.0i)T^{2}
41 1−9.14iT−41T2 1 - 9.14iT - 41T^{2}
43 1+(10.7−2.46i)T+(38.7−18.6i)T2 1 + (10.7 - 2.46i)T + (38.7 - 18.6i)T^{2}
47 1+(3.89−3.10i)T+(10.4−45.8i)T2 1 + (3.89 - 3.10i)T + (10.4 - 45.8i)T^{2}
53 1+(−2.32+10.1i)T+(−47.7−22.9i)T2 1 + (-2.32 + 10.1i)T + (-47.7 - 22.9i)T^{2}
59 1−9.01T+59T2 1 - 9.01T + 59T^{2}
61 1+(−4.16−8.64i)T+(−38.0+47.6i)T2 1 + (-4.16 - 8.64i)T + (-38.0 + 47.6i)T^{2}
67 1+(7.82−9.81i)T+(−14.9−65.3i)T2 1 + (7.82 - 9.81i)T + (-14.9 - 65.3i)T^{2}
71 1+(2.66+3.34i)T+(−15.7+69.2i)T2 1 + (2.66 + 3.34i)T + (-15.7 + 69.2i)T^{2}
73 1+(−6.80+1.55i)T+(65.7−31.6i)T2 1 + (-6.80 + 1.55i)T + (65.7 - 31.6i)T^{2}
79 1+(9.09+7.25i)T+(17.5+77.0i)T2 1 + (9.09 + 7.25i)T + (17.5 + 77.0i)T^{2}
83 1+(−9.76−4.70i)T+(51.7+64.8i)T2 1 + (-9.76 - 4.70i)T + (51.7 + 64.8i)T^{2}
89 1+(2.78+0.634i)T+(80.1+38.6i)T2 1 + (2.78 + 0.634i)T + (80.1 + 38.6i)T^{2}
97 1+(1.05−2.18i)T+(−60.4−75.8i)T2 1 + (1.05 - 2.18i)T + (-60.4 - 75.8i)T^{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

−15.13728038663277387593742098598, −14.73321124051844137234097681855, −12.96523139826998736331195466402, −11.58593935244096194481951383019, −10.29737400718234137986458624291, −9.816370901908045352300578843734, −8.141033604085866630855223395902, −7.15508496456974016707390823056, −4.88175578109012888090872672260, −2.95412219451958159327794785061, 2.02862561492139606548171251717, 4.97360123664242439891011969067, 6.85915834948548848428431638868, 8.081200169120690533548154079707, 8.815452540761104622005400492832, 10.41453237763452535379438099440, 11.74596238501443488128019185687, 12.82052213848433658319168717174, 14.00003912717072950025257867463, 15.15767747313980539985836955715

Graph of the ZZ-function along the critical line