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Properties

Label 2-845-65.37-c1-0-6
Degree 22
Conductor 845845
Sign 0.524−0.851i0.524 - 0.851i
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  + (−0.237 + 0.137i)2-s + (−2.28 − 0.611i)3-s + (−0.962 + 1.66i)4-s + (−1.45 − 1.69i)5-s + (0.627 − 0.168i)6-s + (0.193 − 0.334i)7-s − 1.07i·8-s + (2.23 + 1.29i)9-s + (0.579 + 0.204i)10-s + (−4.21 − 1.12i)11-s + (3.21 − 3.21i)12-s + 0.106i·14-s + (2.27 + 4.76i)15-s + (−1.77 − 3.07i)16-s + (−0.510 − 1.90i)17-s − 0.710·18-s + ⋯
L(s)  = 1  + (−0.168 + 0.0971i)2-s + (−1.31 − 0.353i)3-s + (−0.481 + 0.833i)4-s + (−0.650 − 0.759i)5-s + (0.256 − 0.0686i)6-s + (0.0729 − 0.126i)7-s − 0.381i·8-s + (0.745 + 0.430i)9-s + (0.183 + 0.0646i)10-s + (−1.27 − 0.340i)11-s + (0.928 − 0.928i)12-s + 0.0283i·14-s + (0.588 + 1.23i)15-s + (−0.444 − 0.769i)16-s + (−0.123 − 0.462i)17-s − 0.167·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.524−0.851i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.524−0.851i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.524−0.851i0.524 - 0.851i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(427,⋅)\chi_{845} (427, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 0.524−0.851i)(2,\ 845,\ (\ :1/2),\ 0.524 - 0.851i)

Particular Values

L(1)L(1) ≈\approx 0.254452+0.142169i0.254452 + 0.142169i
L(12)L(\frac12) ≈\approx 0.254452+0.142169i0.254452 + 0.142169i
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+(1.45+1.69i)T 1 + (1.45 + 1.69i)T
13 1 1
good2 1+(0.237−0.137i)T+(1−1.73i)T2 1 + (0.237 - 0.137i)T + (1 - 1.73i)T^{2}
3 1+(2.28+0.611i)T+(2.59+1.5i)T2 1 + (2.28 + 0.611i)T + (2.59 + 1.5i)T^{2}
7 1+(−0.193+0.334i)T+(−3.5−6.06i)T2 1 + (-0.193 + 0.334i)T + (-3.5 - 6.06i)T^{2}
11 1+(4.21+1.12i)T+(9.52+5.5i)T2 1 + (4.21 + 1.12i)T + (9.52 + 5.5i)T^{2}
17 1+(0.510+1.90i)T+(−14.7+8.5i)T2 1 + (0.510 + 1.90i)T + (-14.7 + 8.5i)T^{2}
19 1+(1.29+4.83i)T+(−16.4+9.5i)T2 1 + (1.29 + 4.83i)T + (-16.4 + 9.5i)T^{2}
23 1+(0.0863−0.322i)T+(−19.9−11.5i)T2 1 + (0.0863 - 0.322i)T + (-19.9 - 11.5i)T^{2}
29 1+(7.07−4.08i)T+(14.5−25.1i)T2 1 + (7.07 - 4.08i)T + (14.5 - 25.1i)T^{2}
31 1+(−2.54−2.54i)T+31iT2 1 + (-2.54 - 2.54i)T + 31iT^{2}
37 1+(−2.41−4.17i)T+(−18.5+32.0i)T2 1 + (-2.41 - 4.17i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.20+4.49i)T+(−35.5−20.5i)T2 1 + (-1.20 + 4.49i)T + (-35.5 - 20.5i)T^{2}
43 1+(−6.58+1.76i)T+(37.2−21.5i)T2 1 + (-6.58 + 1.76i)T + (37.2 - 21.5i)T^{2}
47 1−9.83T+47T2 1 - 9.83T + 47T^{2}
53 1+(7.17−7.17i)T−53iT2 1 + (7.17 - 7.17i)T - 53iT^{2}
59 1+(2.34−0.628i)T+(51.0−29.5i)T2 1 + (2.34 - 0.628i)T + (51.0 - 29.5i)T^{2}
61 1+(5.32−9.22i)T+(−30.5−52.8i)T2 1 + (5.32 - 9.22i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5.52+3.18i)T+(33.5−58.0i)T2 1 + (-5.52 + 3.18i)T + (33.5 - 58.0i)T^{2}
71 1+(4.20−1.12i)T+(61.4−35.5i)T2 1 + (4.20 - 1.12i)T + (61.4 - 35.5i)T^{2}
73 1−6.08iT−73T2 1 - 6.08iT - 73T^{2}
79 1+3.34iT−79T2 1 + 3.34iT - 79T^{2}
83 1+5.18T+83T2 1 + 5.18T + 83T^{2}
89 1+(−1.29+4.82i)T+(−77.0−44.5i)T2 1 + (-1.29 + 4.82i)T + (-77.0 - 44.5i)T^{2}
97 1+(−12.7−7.37i)T+(48.5+84.0i)T2 1 + (-12.7 - 7.37i)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

−10.65244152577487062182714819317, −9.254941534997127239608870681345, −8.637886520383381355598555045723, −7.57034740651469986098820559455, −7.18955387632601074076856390338, −5.80943426819131381614536039882, −5.00231808125529356657678931733, −4.30238653434994078869048591137, −2.91275554058756118475190861909, −0.73880104053459999836669802110, 0.28939902591876292627098520729, 2.23418103201486542798163854047, 3.94911025052450711275673623281, 4.79423421993509313363342704532, 5.76657070983033901009771032426, 6.19957294469356536161134163646, 7.51231019763541148413928751738, 8.275803859376179084620414570607, 9.627167097014538101821854948820, 10.27320014481770606892643661250

Graph of the ZZ-function along the critical line