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Properties

Label 2-58-29.11-c2-0-0
Degree 22
Conductor 5858
Sign −0.368−0.929i-0.368 - 0.929i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 + 0.467i)2-s + (0.411 + 3.64i)3-s + (1.56 − 1.24i)4-s + (0.218 − 0.454i)5-s + (−2.25 − 4.67i)6-s + (−7.64 + 9.59i)7-s + (−1.50 + 2.39i)8-s + (−4.37 + 0.997i)9-s + (−0.0799 + 0.709i)10-s + (0.992 + 1.57i)11-s + (5.19 + 5.19i)12-s + (10.1 + 2.30i)13-s + (5.72 − 16.3i)14-s + (1.74 + 0.612i)15-s + (0.890 − 3.89i)16-s + (14.5 − 14.5i)17-s + ⋯
L(s)  = 1  + (−0.667 + 0.233i)2-s + (0.137 + 1.21i)3-s + (0.390 − 0.311i)4-s + (0.0437 − 0.0909i)5-s + (−0.375 − 0.779i)6-s + (−1.09 + 1.37i)7-s + (−0.188 + 0.299i)8-s + (−0.485 + 0.110i)9-s + (−0.00799 + 0.0709i)10-s + (0.0901 + 0.143i)11-s + (0.432 + 0.432i)12-s + (0.777 + 0.177i)13-s + (0.409 − 1.16i)14-s + (0.116 + 0.0408i)15-s + (0.0556 − 0.243i)16-s + (0.858 − 0.858i)17-s + ⋯

Functional equation

Λ(s)=(58s/2 ΓC(s) L(s)=((−0.368−0.929i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(58s/2 ΓC(s+1) L(s)=((−0.368−0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5858    =    2⋅292 \cdot 29
Sign: −0.368−0.929i-0.368 - 0.929i
Analytic conductor: 1.580381.58038
Root analytic conductor: 1.257131.25713
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ58(11,⋅)\chi_{58} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 58, ( :1), −0.368−0.929i)(2,\ 58,\ (\ :1),\ -0.368 - 0.929i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.474062+0.697837i0.474062 + 0.697837i
L(12)L(\frac12) ≈\approx 0.474062+0.697837i0.474062 + 0.697837i
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+(1.33−0.467i)T 1 + (1.33 - 0.467i)T
29 1+(−4.39+28.6i)T 1 + (-4.39 + 28.6i)T
good3 1+(−0.411−3.64i)T+(−8.77+2.00i)T2 1 + (-0.411 - 3.64i)T + (-8.77 + 2.00i)T^{2}
5 1+(−0.218+0.454i)T+(−15.5−19.5i)T2 1 + (-0.218 + 0.454i)T + (-15.5 - 19.5i)T^{2}
7 1+(7.64−9.59i)T+(−10.9−47.7i)T2 1 + (7.64 - 9.59i)T + (-10.9 - 47.7i)T^{2}
11 1+(−0.992−1.57i)T+(−52.4+109.i)T2 1 + (-0.992 - 1.57i)T + (-52.4 + 109. i)T^{2}
13 1+(−10.1−2.30i)T+(152.+73.3i)T2 1 + (-10.1 - 2.30i)T + (152. + 73.3i)T^{2}
17 1+(−14.5+14.5i)T−289iT2 1 + (-14.5 + 14.5i)T - 289iT^{2}
19 1+(9.66+1.08i)T+(351.+80.3i)T2 1 + (9.66 + 1.08i)T + (351. + 80.3i)T^{2}
23 1+(−31.2+15.0i)T+(329.−413.i)T2 1 + (-31.2 + 15.0i)T + (329. - 413. i)T^{2}
31 1+(−3.66+1.28i)T+(751.−599.i)T2 1 + (-3.66 + 1.28i)T + (751. - 599. i)T^{2}
37 1+(28.8−45.9i)T+(−593.−1.23e3i)T2 1 + (28.8 - 45.9i)T + (-593. - 1.23e3i)T^{2}
41 1+(19.7+19.7i)T+1.68e3iT2 1 + (19.7 + 19.7i)T + 1.68e3iT^{2}
43 1+(−13.9+39.8i)T+(−1.44e3−1.15e3i)T2 1 + (-13.9 + 39.8i)T + (-1.44e3 - 1.15e3i)T^{2}
47 1+(55.8−35.0i)T+(958.−1.99e3i)T2 1 + (55.8 - 35.0i)T + (958. - 1.99e3i)T^{2}
53 1+(41.8+20.1i)T+(1.75e3+2.19e3i)T2 1 + (41.8 + 20.1i)T + (1.75e3 + 2.19e3i)T^{2}
59 1−69.3T+3.48e3T2 1 - 69.3T + 3.48e3T^{2}
61 1+(−3.94−35.0i)T+(−3.62e3+828.i)T2 1 + (-3.94 - 35.0i)T + (-3.62e3 + 828. i)T^{2}
67 1+(−111.+25.4i)T+(4.04e3−1.94e3i)T2 1 + (-111. + 25.4i)T + (4.04e3 - 1.94e3i)T^{2}
71 1+(44.2+10.0i)T+(4.54e3+2.18e3i)T2 1 + (44.2 + 10.0i)T + (4.54e3 + 2.18e3i)T^{2}
73 1+(7.08+2.47i)T+(4.16e3+3.32e3i)T2 1 + (7.08 + 2.47i)T + (4.16e3 + 3.32e3i)T^{2}
79 1+(119.+74.8i)T+(2.70e3+5.62e3i)T2 1 + (119. + 74.8i)T + (2.70e3 + 5.62e3i)T^{2}
83 1+(−10.3−12.9i)T+(−1.53e3+6.71e3i)T2 1 + (-10.3 - 12.9i)T + (-1.53e3 + 6.71e3i)T^{2}
89 1+(−141.+49.5i)T+(6.19e3−4.93e3i)T2 1 + (-141. + 49.5i)T + (6.19e3 - 4.93e3i)T^{2}
97 1+(−2.65+23.5i)T+(−9.17e3−2.09e3i)T2 1 + (-2.65 + 23.5i)T + (-9.17e3 - 2.09e3i)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

−15.55901824864154307741345757271, −14.71671184441399882523784226598, −12.96770708547880640656692139746, −11.64069598626368646657282278098, −10.27608566120418118659523343981, −9.353297129068007633300719794662, −8.685022657642728390760700366680, −6.62275860047692305763029445488, −5.19814969627995067249747304040, −3.13298060734985086740534238111, 1.10524545315288457574800217237, 3.43002348639116423745702933750, 6.44435638616473238941529519988, 7.22652889506560312484917960739, 8.446593444369151041667007959328, 10.01671042063103914468371530350, 10.97548056645870480950310121140, 12.65895949092696069094438434789, 13.10886073031330051507393272849, 14.30919901653848782195143251047

Graph of the ZZ-function along the critical line