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Properties

Label 2-58-29.7-c1-0-1
Degree 22
Conductor 5858
Sign 0.895+0.444i0.895 + 0.444i
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  + (−0.222 − 0.974i)2-s + (1.56 + 0.755i)3-s + (−0.900 + 0.433i)4-s + (0.110 + 0.482i)5-s + (0.387 − 1.69i)6-s + (−2.82 − 1.36i)7-s + (0.623 + 0.781i)8-s + (0.0208 + 0.0261i)9-s + (0.445 − 0.214i)10-s + (0.870 − 1.09i)11-s − 1.74·12-s + (−3.56 + 4.46i)13-s + (−0.698 + 3.05i)14-s + (−0.191 + 0.840i)15-s + (0.623 − 0.781i)16-s + 5.31·17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (0.905 + 0.436i)3-s + (−0.450 + 0.216i)4-s + (0.0492 + 0.215i)5-s + (0.158 − 0.693i)6-s + (−1.06 − 0.514i)7-s + (0.220 + 0.276i)8-s + (0.00695 + 0.00871i)9-s + (0.140 − 0.0678i)10-s + (0.262 − 0.329i)11-s − 0.502·12-s + (−0.988 + 1.23i)13-s + (−0.186 + 0.817i)14-s + (−0.0495 + 0.216i)15-s + (0.155 − 0.195i)16-s + 1.28·17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((0.895+0.444i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1/2) L(s)=((0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: 0.895+0.444i0.895 + 0.444i
Analytic conductor: 0.4631320.463132
Root analytic conductor: 0.6805380.680538
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ58(7,⋅)\chi_{58} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1/2), 0.895+0.444i)(2,\ 58,\ (\ :1/2),\ 0.895 + 0.444i)

Particular Values

L(1)L(1) ≈\approx 0.888209−0.208146i0.888209 - 0.208146i
L(12)L(\frac12) ≈\approx 0.888209−0.208146i0.888209 - 0.208146i
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+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
29 1+(−5.38−0.0414i)T 1 + (-5.38 - 0.0414i)T
good3 1+(−1.56−0.755i)T+(1.87+2.34i)T2 1 + (-1.56 - 0.755i)T + (1.87 + 2.34i)T^{2}
5 1+(−0.110−0.482i)T+(−4.50+2.16i)T2 1 + (-0.110 - 0.482i)T + (-4.50 + 2.16i)T^{2}
7 1+(2.82+1.36i)T+(4.36+5.47i)T2 1 + (2.82 + 1.36i)T + (4.36 + 5.47i)T^{2}
11 1+(−0.870+1.09i)T+(−2.44−10.7i)T2 1 + (-0.870 + 1.09i)T + (-2.44 - 10.7i)T^{2}
13 1+(3.56−4.46i)T+(−2.89−12.6i)T2 1 + (3.56 - 4.46i)T + (-2.89 - 12.6i)T^{2}
17 1−5.31T+17T2 1 - 5.31T + 17T^{2}
19 1+(4.47−2.15i)T+(11.8−14.8i)T2 1 + (4.47 - 2.15i)T + (11.8 - 14.8i)T^{2}
23 1+(−0.181+0.793i)T+(−20.7−9.97i)T2 1 + (-0.181 + 0.793i)T + (-20.7 - 9.97i)T^{2}
31 1+(1.41+6.20i)T+(−27.9+13.4i)T2 1 + (1.41 + 6.20i)T + (-27.9 + 13.4i)T^{2}
37 1+(−5.56−6.97i)T+(−8.23+36.0i)T2 1 + (-5.56 - 6.97i)T + (-8.23 + 36.0i)T^{2}
41 1+4.01T+41T2 1 + 4.01T + 41T^{2}
43 1+(−0.310+1.36i)T+(−38.7−18.6i)T2 1 + (-0.310 + 1.36i)T + (-38.7 - 18.6i)T^{2}
47 1+(6.42−8.05i)T+(−10.4−45.8i)T2 1 + (6.42 - 8.05i)T + (-10.4 - 45.8i)T^{2}
53 1+(0.944+4.13i)T+(−47.7+22.9i)T2 1 + (0.944 + 4.13i)T + (-47.7 + 22.9i)T^{2}
59 1−11.1T+59T2 1 - 11.1T + 59T^{2}
61 1+(4.38+2.11i)T+(38.0+47.6i)T2 1 + (4.38 + 2.11i)T + (38.0 + 47.6i)T^{2}
67 1+(−4.45−5.58i)T+(−14.9+65.3i)T2 1 + (-4.45 - 5.58i)T + (-14.9 + 65.3i)T^{2}
71 1+(−3.76+4.72i)T+(−15.7−69.2i)T2 1 + (-3.76 + 4.72i)T + (-15.7 - 69.2i)T^{2}
73 1+(2.73−11.9i)T+(−65.7−31.6i)T2 1 + (2.73 - 11.9i)T + (-65.7 - 31.6i)T^{2}
79 1+(5.86+7.35i)T+(−17.5+77.0i)T2 1 + (5.86 + 7.35i)T + (-17.5 + 77.0i)T^{2}
83 1+(11.4−5.51i)T+(51.7−64.8i)T2 1 + (11.4 - 5.51i)T + (51.7 - 64.8i)T^{2}
89 1+(0.398+1.74i)T+(−80.1+38.6i)T2 1 + (0.398 + 1.74i)T + (-80.1 + 38.6i)T^{2}
97 1+(3.42−1.64i)T+(60.4−75.8i)T2 1 + (3.42 - 1.64i)T + (60.4 - 75.8i)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

−14.71496146257308266435565361037, −14.14561189836495382902830381805, −12.86781172681557728591155802493, −11.72650993109979044333554479108, −10.11306504593307121714581040135, −9.564520212765823918553394556079, −8.290700106888424980790488671685, −6.58664819165781313818070734103, −4.14835157466600069837477845827, −2.88304899228522888301033541176, 2.97289032003055873712764771288, 5.34179659423460159677096180889, 6.91261036887911701110291974698, 8.092593397426579649064035243793, 9.142357688520052268455155309236, 10.24135360093120599801891144439, 12.46742317437420017130985695710, 13.04591739612882840996438901497, 14.41780360727538971274792358777, 15.09727535894556795918395182625

Graph of the ZZ-function along the critical line