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Properties

Label 2-845-65.37-c1-0-52
Degree 22
Conductor 845845
Sign 0.851+0.523i0.851 + 0.523i
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  + (−1.58 + 0.915i)2-s + (1.91 + 0.512i)3-s + (0.677 − 1.17i)4-s + (1.69 − 1.45i)5-s + (−3.50 + 0.939i)6-s + (1.76 − 3.06i)7-s − 1.18i·8-s + (0.803 + 0.463i)9-s + (−1.36 + 3.86i)10-s + (−3.74 − 1.00i)11-s + (1.89 − 1.89i)12-s + 6.48i·14-s + (3.99 − 1.91i)15-s + (2.43 + 4.22i)16-s + (−0.524 − 1.95i)17-s − 1.69·18-s + ⋯
L(s)  = 1  + (−1.12 + 0.647i)2-s + (1.10 + 0.296i)3-s + (0.338 − 0.586i)4-s + (0.759 − 0.650i)5-s + (−1.43 + 0.383i)6-s + (0.668 − 1.15i)7-s − 0.417i·8-s + (0.267 + 0.154i)9-s + (−0.430 + 1.22i)10-s + (−1.12 − 0.302i)11-s + (0.548 − 0.548i)12-s + 1.73i·14-s + (1.03 − 0.494i)15-s + (0.609 + 1.05i)16-s + (−0.127 − 0.474i)17-s − 0.400·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.851+0.523i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.851+0.523i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.851+0.523i0.851 + 0.523i
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), 0.851+0.523i)(2,\ 845,\ (\ :1/2),\ 0.851 + 0.523i)

Particular Values

L(1)L(1) ≈\approx 1.22895−0.347584i1.22895 - 0.347584i
L(12)L(\frac12) ≈\approx 1.22895−0.347584i1.22895 - 0.347584i
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+(−1.69+1.45i)T 1 + (-1.69 + 1.45i)T
13 1 1
good2 1+(1.58−0.915i)T+(1−1.73i)T2 1 + (1.58 - 0.915i)T + (1 - 1.73i)T^{2}
3 1+(−1.91−0.512i)T+(2.59+1.5i)T2 1 + (-1.91 - 0.512i)T + (2.59 + 1.5i)T^{2}
7 1+(−1.76+3.06i)T+(−3.5−6.06i)T2 1 + (-1.76 + 3.06i)T + (-3.5 - 6.06i)T^{2}
11 1+(3.74+1.00i)T+(9.52+5.5i)T2 1 + (3.74 + 1.00i)T + (9.52 + 5.5i)T^{2}
17 1+(0.524+1.95i)T+(−14.7+8.5i)T2 1 + (0.524 + 1.95i)T + (-14.7 + 8.5i)T^{2}
19 1+(−0.139−0.518i)T+(−16.4+9.5i)T2 1 + (-0.139 - 0.518i)T + (-16.4 + 9.5i)T^{2}
23 1+(0.0788−0.294i)T+(−19.9−11.5i)T2 1 + (0.0788 - 0.294i)T + (-19.9 - 11.5i)T^{2}
29 1+(−1.71+0.988i)T+(14.5−25.1i)T2 1 + (-1.71 + 0.988i)T + (14.5 - 25.1i)T^{2}
31 1+(4.13+4.13i)T+31iT2 1 + (4.13 + 4.13i)T + 31iT^{2}
37 1+(−2.70−4.69i)T+(−18.5+32.0i)T2 1 + (-2.70 - 4.69i)T + (-18.5 + 32.0i)T^{2}
41 1+(−0.174+0.649i)T+(−35.5−20.5i)T2 1 + (-0.174 + 0.649i)T + (-35.5 - 20.5i)T^{2}
43 1+(−8.51+2.28i)T+(37.2−21.5i)T2 1 + (-8.51 + 2.28i)T + (37.2 - 21.5i)T^{2}
47 1+9.75T+47T2 1 + 9.75T + 47T^{2}
53 1+(−3.16+3.16i)T−53iT2 1 + (-3.16 + 3.16i)T - 53iT^{2}
59 1+(−11.7+3.14i)T+(51.0−29.5i)T2 1 + (-11.7 + 3.14i)T + (51.0 - 29.5i)T^{2}
61 1+(−1.44+2.49i)T+(−30.5−52.8i)T2 1 + (-1.44 + 2.49i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.98−1.14i)T+(33.5−58.0i)T2 1 + (1.98 - 1.14i)T + (33.5 - 58.0i)T^{2}
71 1+(4.46−1.19i)T+(61.4−35.5i)T2 1 + (4.46 - 1.19i)T + (61.4 - 35.5i)T^{2}
73 1−14.7iT−73T2 1 - 14.7iT - 73T^{2}
79 1−1.59iT−79T2 1 - 1.59iT - 79T^{2}
83 1−7.57T+83T2 1 - 7.57T + 83T^{2}
89 1+(−1.21+4.54i)T+(−77.0−44.5i)T2 1 + (-1.21 + 4.54i)T + (-77.0 - 44.5i)T^{2}
97 1+(−15.4−8.91i)T+(48.5+84.0i)T2 1 + (-15.4 - 8.91i)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

−9.880261727257276874000286191878, −9.145167622144313311158439701952, −8.348368281356304747805046868696, −7.923213358441146925355760423879, −7.13067014532073337970704141955, −5.90160403349277794072128335153, −4.74609962024195639579653752908, −3.67718330334952368975448560202, −2.26889023441531115661583985077, −0.78202258286069750546902579239, 1.81124811755786337625005788185, 2.34896420907734707760125727368, 3.08321452138816408287915723923, 5.08154423393561238569679262911, 5.87440373697325364717696175747, 7.35881034021701452257061666567, 8.015254528508575538502252166260, 8.802553062409282208904417909726, 9.227669281035982340793267457223, 10.19468967609923140447885734601

Graph of the ZZ-function along the critical line