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Properties

Label 2-845-65.58-c1-0-26
Degree 22
Conductor 845845
Sign −0.803+0.594i-0.803 + 0.594i
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  + (−2.29 − 1.32i)2-s + (−1.25 + 0.335i)3-s + (2.51 + 4.34i)4-s + (−1.81 − 1.30i)5-s + (3.31 + 0.889i)6-s + (−0.0561 − 0.0972i)7-s − 8.00i·8-s + (−1.14 + 0.658i)9-s + (2.44 + 5.39i)10-s + (1.78 − 0.479i)11-s + (−4.60 − 4.60i)12-s + 0.297i·14-s + (2.71 + 1.02i)15-s + (−5.58 + 9.67i)16-s + (0.706 − 2.63i)17-s + 3.49·18-s + ⋯
L(s)  = 1  + (−1.62 − 0.936i)2-s + (−0.723 + 0.193i)3-s + (1.25 + 2.17i)4-s + (−0.812 − 0.583i)5-s + (1.35 + 0.363i)6-s + (−0.0212 − 0.0367i)7-s − 2.83i·8-s + (−0.380 + 0.219i)9-s + (0.771 + 1.70i)10-s + (0.539 − 0.144i)11-s + (−1.32 − 1.32i)12-s + 0.0795i·14-s + (0.700 + 0.264i)15-s + (−1.39 + 2.41i)16-s + (0.171 − 0.639i)17-s + 0.823·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.803+0.594i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.803+0.594i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.803+0.594i-0.803 + 0.594i
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), −0.803+0.594i)(2,\ 845,\ (\ :1/2),\ -0.803 + 0.594i)

Particular Values

L(1)L(1) ≈\approx 0.0743354−0.225419i0.0743354 - 0.225419i
L(12)L(\frac12) ≈\approx 0.0743354−0.225419i0.0743354 - 0.225419i
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+(1.81+1.30i)T 1 + (1.81 + 1.30i)T
13 1 1
good2 1+(2.29+1.32i)T+(1+1.73i)T2 1 + (2.29 + 1.32i)T + (1 + 1.73i)T^{2}
3 1+(1.25−0.335i)T+(2.59−1.5i)T2 1 + (1.25 - 0.335i)T + (2.59 - 1.5i)T^{2}
7 1+(0.0561+0.0972i)T+(−3.5+6.06i)T2 1 + (0.0561 + 0.0972i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.78+0.479i)T+(9.52−5.5i)T2 1 + (-1.78 + 0.479i)T + (9.52 - 5.5i)T^{2}
17 1+(−0.706+2.63i)T+(−14.7−8.5i)T2 1 + (-0.706 + 2.63i)T + (-14.7 - 8.5i)T^{2}
19 1+(1.80−6.72i)T+(−16.4−9.5i)T2 1 + (1.80 - 6.72i)T + (-16.4 - 9.5i)T^{2}
23 1+(−0.831−3.10i)T+(−19.9+11.5i)T2 1 + (-0.831 - 3.10i)T + (-19.9 + 11.5i)T^{2}
29 1+(−4.03−2.32i)T+(14.5+25.1i)T2 1 + (-4.03 - 2.32i)T + (14.5 + 25.1i)T^{2}
31 1+(−0.624+0.624i)T−31iT2 1 + (-0.624 + 0.624i)T - 31iT^{2}
37 1+(−0.737+1.27i)T+(−18.5−32.0i)T2 1 + (-0.737 + 1.27i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.40+5.24i)T+(−35.5+20.5i)T2 1 + (1.40 + 5.24i)T + (-35.5 + 20.5i)T^{2}
43 1+(3.76+1.00i)T+(37.2+21.5i)T2 1 + (3.76 + 1.00i)T + (37.2 + 21.5i)T^{2}
47 1−0.345T+47T2 1 - 0.345T + 47T^{2}
53 1+(3.59+3.59i)T+53iT2 1 + (3.59 + 3.59i)T + 53iT^{2}
59 1+(−1.24−0.332i)T+(51.0+29.5i)T2 1 + (-1.24 - 0.332i)T + (51.0 + 29.5i)T^{2}
61 1+(−1.39−2.41i)T+(−30.5+52.8i)T2 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.124−0.0721i)T+(33.5+58.0i)T2 1 + (-0.124 - 0.0721i)T + (33.5 + 58.0i)T^{2}
71 1+(5.28+1.41i)T+(61.4+35.5i)T2 1 + (5.28 + 1.41i)T + (61.4 + 35.5i)T^{2}
73 1+9.06iT−73T2 1 + 9.06iT - 73T^{2}
79 1+15.1iT−79T2 1 + 15.1iT - 79T^{2}
83 1+8.53T+83T2 1 + 8.53T + 83T^{2}
89 1+(−0.147−0.549i)T+(−77.0+44.5i)T2 1 + (-0.147 - 0.549i)T + (-77.0 + 44.5i)T^{2}
97 1+(12.9−7.48i)T+(48.5−84.0i)T2 1 + (12.9 - 7.48i)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

−9.969190197398864705494306802198, −9.030776627345528831745450274921, −8.381809411198200370748457167914, −7.70856205524369122945371390272, −6.73281974203889649946178523494, −5.43661717999672247662189267321, −4.10181469932528343446052332121, −3.12172882051357462708294590942, −1.57582273888687760049923518933, −0.29623316250245102333529795452, 0.951536001703881398986142017293, 2.73784738597263207353321521792, 4.54144968521801097750621925681, 5.81199198462306149406434553741, 6.68930639299355697665711437712, 6.90631009232848781580808091516, 8.107653756811752401095042200683, 8.626201387702127005217789949517, 9.544737143642516666363667291324, 10.49260401769523422336242428273

Graph of the ZZ-function along the critical line