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Properties

Label 2-58-29.13-c1-0-0
Degree 22
Conductor 5858
Sign 0.471−0.881i0.471 - 0.881i
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  + (−0.433 + 0.900i)2-s + (2.19 + 1.74i)3-s + (−0.623 − 0.781i)4-s + (−3.43 − 1.65i)5-s + (−2.52 + 1.21i)6-s + (1.36 − 1.71i)7-s + (0.974 − 0.222i)8-s + (1.07 + 4.72i)9-s + (2.98 − 2.38i)10-s + (−0.0647 − 0.0147i)11-s − 2.80i·12-s + (−0.157 + 0.687i)13-s + (0.949 + 1.97i)14-s + (−4.64 − 9.63i)15-s + (−0.222 + 0.974i)16-s − 3.46i·17-s + ⋯
L(s)  = 1  + (−0.306 + 0.637i)2-s + (1.26 + 1.00i)3-s + (−0.311 − 0.390i)4-s + (−1.53 − 0.740i)5-s + (−1.03 + 0.496i)6-s + (0.515 − 0.646i)7-s + (0.344 − 0.0786i)8-s + (0.359 + 1.57i)9-s + (0.943 − 0.752i)10-s + (−0.0195 − 0.00445i)11-s − 0.808i·12-s + (−0.0435 + 0.190i)13-s + (0.253 + 0.526i)14-s + (−1.19 − 2.48i)15-s + (−0.0556 + 0.243i)16-s − 0.839i·17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((0.471−0.881i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1/2) L(s)=((0.471−0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: 0.471−0.881i0.471 - 0.881i
Analytic conductor: 0.4631320.463132
Root analytic conductor: 0.6805380.680538
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ58(13,⋅)\chi_{58} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1/2), 0.471−0.881i)(2,\ 58,\ (\ :1/2),\ 0.471 - 0.881i)

Particular Values

L(1)L(1) ≈\approx 0.755380+0.452608i0.755380 + 0.452608i
L(12)L(\frac12) ≈\approx 0.755380+0.452608i0.755380 + 0.452608i
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+(0.433−0.900i)T 1 + (0.433 - 0.900i)T
29 1+(−2.16−4.92i)T 1 + (-2.16 - 4.92i)T
good3 1+(−2.19−1.74i)T+(0.667+2.92i)T2 1 + (-2.19 - 1.74i)T + (0.667 + 2.92i)T^{2}
5 1+(3.43+1.65i)T+(3.11+3.90i)T2 1 + (3.43 + 1.65i)T + (3.11 + 3.90i)T^{2}
7 1+(−1.36+1.71i)T+(−1.55−6.82i)T2 1 + (-1.36 + 1.71i)T + (-1.55 - 6.82i)T^{2}
11 1+(0.0647+0.0147i)T+(9.91+4.77i)T2 1 + (0.0647 + 0.0147i)T + (9.91 + 4.77i)T^{2}
13 1+(0.157−0.687i)T+(−11.7−5.64i)T2 1 + (0.157 - 0.687i)T + (-11.7 - 5.64i)T^{2}
17 1+3.46iT−17T2 1 + 3.46iT - 17T^{2}
19 1+(2.15−1.72i)T+(4.22−18.5i)T2 1 + (2.15 - 1.72i)T + (4.22 - 18.5i)T^{2}
23 1+(5.68−2.73i)T+(14.3−17.9i)T2 1 + (5.68 - 2.73i)T + (14.3 - 17.9i)T^{2}
31 1+(−3.24+6.74i)T+(−19.3−24.2i)T2 1 + (-3.24 + 6.74i)T + (-19.3 - 24.2i)T^{2}
37 1+(8.31−1.89i)T+(33.3−16.0i)T2 1 + (8.31 - 1.89i)T + (33.3 - 16.0i)T^{2}
41 1−2.48iT−41T2 1 - 2.48iT - 41T^{2}
43 1+(0.624+1.29i)T+(−26.8+33.6i)T2 1 + (0.624 + 1.29i)T + (-26.8 + 33.6i)T^{2}
47 1+(−8.77−2.00i)T+(42.3+20.3i)T2 1 + (-8.77 - 2.00i)T + (42.3 + 20.3i)T^{2}
53 1+(−2.90−1.40i)T+(33.0+41.4i)T2 1 + (-2.90 - 1.40i)T + (33.0 + 41.4i)T^{2}
59 1−1.24T+59T2 1 - 1.24T + 59T^{2}
61 1+(−2.61−2.08i)T+(13.5+59.4i)T2 1 + (-2.61 - 2.08i)T + (13.5 + 59.4i)T^{2}
67 1+(1.56+6.83i)T+(−60.3+29.0i)T2 1 + (1.56 + 6.83i)T + (-60.3 + 29.0i)T^{2}
71 1+(−2.24+9.84i)T+(−63.9−30.8i)T2 1 + (-2.24 + 9.84i)T + (-63.9 - 30.8i)T^{2}
73 1+(−1.31−2.73i)T+(−45.5+57.0i)T2 1 + (-1.31 - 2.73i)T + (-45.5 + 57.0i)T^{2}
79 1+(−2.17+0.497i)T+(71.1−34.2i)T2 1 + (-2.17 + 0.497i)T + (71.1 - 34.2i)T^{2}
83 1+(3.82+4.79i)T+(−18.4+80.9i)T2 1 + (3.82 + 4.79i)T + (-18.4 + 80.9i)T^{2}
89 1+(−7.59+15.7i)T+(−55.4−69.5i)T2 1 + (-7.59 + 15.7i)T + (-55.4 - 69.5i)T^{2}
97 1+(1.72−1.37i)T+(21.5−94.5i)T2 1 + (1.72 - 1.37i)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

−15.56627221977377300087634536906, −14.56332265420272088652104753752, −13.66238402314968807641065588515, −11.92368776132151169313305339790, −10.48689436724372215945374906664, −9.174340012208870987245927808779, −8.230017133506104101219716494483, −7.52660254300522152994342854592, −4.71804452420580215597278671236, −3.83227926987666197740733034857, 2.43973740234156098959406793371, 3.85089157060427079795415542081, 6.95558415390046699058583900476, 8.156113741894002568085331732779, 8.550469245682265001244106076945, 10.52304575358119410725165112695, 11.86679795551209202203146726032, 12.49037408651506710036756494636, 13.93121470601692306879404371545, 14.86224018941700166606202162060

Graph of the ZZ-function along the critical line