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Properties

Label 2-845-65.58-c1-0-61
Degree 22
Conductor 845845
Sign 0.756+0.653i0.756 + 0.653i
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)=((0.756+0.653i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.756+0.653i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.756+0.653i0.756 + 0.653i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(188,⋅)\chi_{845} (188, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 0.756+0.653i)(2,\ 845,\ (\ :1/2),\ 0.756 + 0.653i)

Particular Values

L(1)L(1) ≈\approx 3.03108−1.12807i3.03108 - 1.12807i
L(12)L(\frac12) ≈\approx 3.03108−1.12807i3.03108 - 1.12807i
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

−9.886259563878319325645835204975, −9.063581127621240230940540996530, −8.214962736404883627961236136390, −7.38069128185452210406845807205, −6.73550577767773583389486207404, −5.73966325266268790683145334903, −4.31260568747681400707897549377, −3.95448044125985351484206889874, −3.06571671996911120790685882717, −1.04703815158443639442378688784, 2.30260352543598557120806378710, 2.95849149311839495224620385358, 3.69752546918475129068920871830, 4.46114918038890962847188240163, 5.77300806382504610724631189256, 6.69556194470941698557382213678, 7.85013132681904359255727915539, 8.841253200348573528534384412662, 9.255340441487114465941450446554, 10.42780698842025180818387344678

Graph of the ZZ-function along the critical line