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Properties

Label 2-845-65.7-c1-0-35
Degree 22
Conductor 845845
Sign −0.293+0.955i-0.293 + 0.955i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
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  + (−1.94 − 1.12i)2-s + (0.514 + 1.91i)3-s + (1.53 + 2.65i)4-s + (−0.247 + 2.22i)5-s + (1.15 − 4.31i)6-s + (−0.638 − 1.10i)7-s − 2.39i·8-s + (−0.820 + 0.473i)9-s + (2.98 − 4.05i)10-s + (−1.41 − 5.27i)11-s + (−4.30 + 4.30i)12-s + 2.87i·14-s + (−4.39 + 0.666i)15-s + (0.365 − 0.633i)16-s + (−3.11 − 0.833i)17-s + 2.13·18-s + ⋯
L(s)  = 1  + (−1.37 − 0.795i)2-s + (0.296 + 1.10i)3-s + (0.766 + 1.32i)4-s + (−0.110 + 0.993i)5-s + (0.472 − 1.76i)6-s + (−0.241 − 0.418i)7-s − 0.848i·8-s + (−0.273 + 0.157i)9-s + (0.943 − 1.28i)10-s + (−0.426 − 1.59i)11-s + (−1.24 + 1.24i)12-s + 0.768i·14-s + (−1.13 + 0.172i)15-s + (0.0913 − 0.158i)16-s + (−0.754 − 0.202i)17-s + 0.502·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.293+0.955i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.293+0.955i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.293+0.955i-0.293 + 0.955i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(657,⋅)\chi_{845} (657, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), −0.293+0.955i)(2,\ 845,\ (\ :1/2),\ -0.293 + 0.955i)

Particular Values

L(1)L(1) ≈\approx 0.217193−0.293889i0.217193 - 0.293889i
L(12)L(\frac12) ≈\approx 0.217193−0.293889i0.217193 - 0.293889i
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)
bad5 1+(0.247−2.22i)T 1 + (0.247 - 2.22i)T
13 1 1
good2 1+(1.94+1.12i)T+(1+1.73i)T2 1 + (1.94 + 1.12i)T + (1 + 1.73i)T^{2}
3 1+(−0.514−1.91i)T+(−2.59+1.5i)T2 1 + (-0.514 - 1.91i)T + (-2.59 + 1.5i)T^{2}
7 1+(0.638+1.10i)T+(−3.5+6.06i)T2 1 + (0.638 + 1.10i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.41+5.27i)T+(−9.52+5.5i)T2 1 + (1.41 + 5.27i)T + (-9.52 + 5.5i)T^{2}
17 1+(3.11+0.833i)T+(14.7+8.5i)T2 1 + (3.11 + 0.833i)T + (14.7 + 8.5i)T^{2}
19 1+(1.17+0.315i)T+(16.4+9.5i)T2 1 + (1.17 + 0.315i)T + (16.4 + 9.5i)T^{2}
23 1+(0.160−0.0428i)T+(19.9−11.5i)T2 1 + (0.160 - 0.0428i)T + (19.9 - 11.5i)T^{2}
29 1+(8.41+4.85i)T+(14.5+25.1i)T2 1 + (8.41 + 4.85i)T + (14.5 + 25.1i)T^{2}
31 1+(0.233+0.233i)T+31iT2 1 + (0.233 + 0.233i)T + 31iT^{2}
37 1+(−0.660+1.14i)T+(−18.5−32.0i)T2 1 + (-0.660 + 1.14i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.483+0.129i)T+(35.5−20.5i)T2 1 + (-0.483 + 0.129i)T + (35.5 - 20.5i)T^{2}
43 1+(−1.72+6.43i)T+(−37.2−21.5i)T2 1 + (-1.72 + 6.43i)T + (-37.2 - 21.5i)T^{2}
47 1+3.20T+47T2 1 + 3.20T + 47T^{2}
53 1+(−4.49+4.49i)T−53iT2 1 + (-4.49 + 4.49i)T - 53iT^{2}
59 1+(−0.000595+0.00222i)T+(−51.0−29.5i)T2 1 + (-0.000595 + 0.00222i)T + (-51.0 - 29.5i)T^{2}
61 1+(0.695+1.20i)T+(−30.5+52.8i)T2 1 + (0.695 + 1.20i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.26−3.03i)T+(33.5+58.0i)T2 1 + (-5.26 - 3.03i)T + (33.5 + 58.0i)T^{2}
71 1+(−3.14+11.7i)T+(−61.4−35.5i)T2 1 + (-3.14 + 11.7i)T + (-61.4 - 35.5i)T^{2}
73 1−7.34iT−73T2 1 - 7.34iT - 73T^{2}
79 1+11.1iT−79T2 1 + 11.1iT - 79T^{2}
83 1+2.65T+83T2 1 + 2.65T + 83T^{2}
89 1+(6.96−1.86i)T+(77.0−44.5i)T2 1 + (6.96 - 1.86i)T + (77.0 - 44.5i)T^{2}
97 1+(3.62−2.09i)T+(48.5−84.0i)T2 1 + (3.62 - 2.09i)T + (48.5 - 84.0i)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

−10.01251702688155489218448275327, −9.299092363968461692270821388831, −8.586643420876632945907413585915, −7.74063486849168192149127964859, −6.77847173187448542775312695256, −5.54902641259005424173502733689, −3.94188682090598800179604077578, −3.29693331470196501464841157617, −2.30251834534769965002577652905, −0.27557898599933907614561398826, 1.37713461139747705644991336144, 2.21403134661476178188291967679, 4.32633115614740930470078915050, 5.52150594868061906355404120694, 6.58458989413385604479690669943, 7.30154697578829511933577890685, 7.86445385467574321835213280987, 8.632626364842660819250316688346, 9.325731565516576870870267198800, 9.973190533523095819997104061721

Graph of the ZZ-function along the critical line