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Properties

Label 2-845-65.7-c1-0-41
Degree 22
Conductor 845845
Sign 0.940+0.340i0.940 + 0.340i
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 + (−0.512 − 1.91i)3-s + (0.677 + 1.17i)4-s + (1.69 − 1.45i)5-s + (0.939 − 3.50i)6-s + (1.76 + 3.06i)7-s − 1.18i·8-s + (−0.803 + 0.463i)9-s + (4.02 − 0.752i)10-s + (1.00 + 3.74i)11-s + (1.89 − 1.89i)12-s + 6.48i·14-s + (−3.65 − 2.50i)15-s + (2.43 − 4.22i)16-s + (1.95 + 0.524i)17-s − 1.69·18-s + ⋯
L(s)  = 1  + (1.12 + 0.647i)2-s + (−0.296 − 1.10i)3-s + (0.338 + 0.586i)4-s + (0.759 − 0.650i)5-s + (0.383 − 1.43i)6-s + (0.668 + 1.15i)7-s − 0.417i·8-s + (−0.267 + 0.154i)9-s + (1.27 − 0.237i)10-s + (0.302 + 1.12i)11-s + (0.548 − 0.548i)12-s + 1.73i·14-s + (−0.943 − 0.646i)15-s + (0.609 − 1.05i)16-s + (0.474 + 0.127i)17-s − 0.400·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.940+0.340i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.940+0.340i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.940+0.340i0.940 + 0.340i
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), 0.940+0.340i)(2,\ 845,\ (\ :1/2),\ 0.940 + 0.340i)

Particular Values

L(1)L(1) ≈\approx 3.06915−0.538620i3.06915 - 0.538620i
L(12)L(\frac12) ≈\approx 3.06915−0.538620i3.06915 - 0.538620i
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+(0.512+1.91i)T+(−2.59+1.5i)T2 1 + (0.512 + 1.91i)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+(−1.00−3.74i)T+(−9.52+5.5i)T2 1 + (-1.00 - 3.74i)T + (-9.52 + 5.5i)T^{2}
17 1+(−1.95−0.524i)T+(14.7+8.5i)T2 1 + (-1.95 - 0.524i)T + (14.7 + 8.5i)T^{2}
19 1+(0.518+0.139i)T+(16.4+9.5i)T2 1 + (0.518 + 0.139i)T + (16.4 + 9.5i)T^{2}
23 1+(−0.294+0.0788i)T+(19.9−11.5i)T2 1 + (-0.294 + 0.0788i)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+(0.649−0.174i)T+(35.5−20.5i)T2 1 + (0.649 - 0.174i)T + (35.5 - 20.5i)T^{2}
43 1+(2.28−8.51i)T+(−37.2−21.5i)T2 1 + (2.28 - 8.51i)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+(3.14−11.7i)T+(−51.0−29.5i)T2 1 + (3.14 - 11.7i)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+(−1.19+4.46i)T+(−61.4−35.5i)T2 1 + (-1.19 + 4.46i)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+(4.54−1.21i)T+(77.0−44.5i)T2 1 + (4.54 - 1.21i)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

−9.924348652873880090788632466781, −9.297870130402568829817245081540, −8.152249991945685832435715572816, −7.30782033652077956986638654745, −6.41866343392385829446982987202, −5.75024934838796631770986845746, −5.10935050059026889285046138926, −4.17654878715663322820303428707, −2.35253488640686339521977504910, −1.39612149501692483280436928722, 1.66455531984593899426296289389, 3.25540819336390688373578736699, 3.75365766452158288098844516485, 4.81432045763821304972414176585, 5.40673166167496895113815774821, 6.41950503248978605406313294484, 7.60039841154012086062307504990, 8.739944427890165704839592146589, 9.864132021730974351564906183112, 10.53261787771275877583899657120

Graph of the ZZ-function along the critical line