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Properties

Label 2-845-65.7-c1-0-61
Degree 22
Conductor 845845
Sign −0.847−0.530i-0.847 - 0.530i
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  + (−0.113 − 0.0656i)2-s + (−0.0890 − 0.332i)3-s + (−0.991 − 1.71i)4-s + (−2.08 − 0.813i)5-s + (−0.0116 + 0.0436i)6-s + (−1.39 − 2.40i)7-s + 0.522i·8-s + (2.49 − 1.44i)9-s + (0.183 + 0.229i)10-s + (−1.04 − 3.91i)11-s + (−0.482 + 0.482i)12-s + 0.365i·14-s + (−0.0847 + 0.764i)15-s + (−1.94 + 3.37i)16-s + (2.34 + 0.627i)17-s − 0.378·18-s + ⋯
L(s)  = 1  + (−0.0804 − 0.0464i)2-s + (−0.0513 − 0.191i)3-s + (−0.495 − 0.858i)4-s + (−0.931 − 0.363i)5-s + (−0.00477 + 0.0178i)6-s + (−0.525 − 0.910i)7-s + 0.184i·8-s + (0.831 − 0.480i)9-s + (0.0580 + 0.0724i)10-s + (−0.316 − 1.18i)11-s + (−0.139 + 0.139i)12-s + 0.0976i·14-s + (−0.0218 + 0.197i)15-s + (−0.487 + 0.843i)16-s + (0.567 + 0.152i)17-s − 0.0891·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((−0.847−0.530i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((−0.847−0.530i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: −0.847−0.530i-0.847 - 0.530i
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), −0.847−0.530i)(2,\ 845,\ (\ :1/2),\ -0.847 - 0.530i)

Particular Values

L(1)L(1) ≈\approx 0.143347+0.499454i0.143347 + 0.499454i
L(12)L(\frac12) ≈\approx 0.143347+0.499454i0.143347 + 0.499454i
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+(2.08+0.813i)T 1 + (2.08 + 0.813i)T
13 1 1
good2 1+(0.113+0.0656i)T+(1+1.73i)T2 1 + (0.113 + 0.0656i)T + (1 + 1.73i)T^{2}
3 1+(0.0890+0.332i)T+(−2.59+1.5i)T2 1 + (0.0890 + 0.332i)T + (-2.59 + 1.5i)T^{2}
7 1+(1.39+2.40i)T+(−3.5+6.06i)T2 1 + (1.39 + 2.40i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.04+3.91i)T+(−9.52+5.5i)T2 1 + (1.04 + 3.91i)T + (-9.52 + 5.5i)T^{2}
17 1+(−2.34−0.627i)T+(14.7+8.5i)T2 1 + (-2.34 - 0.627i)T + (14.7 + 8.5i)T^{2}
19 1+(−1.83−0.491i)T+(16.4+9.5i)T2 1 + (-1.83 - 0.491i)T + (16.4 + 9.5i)T^{2}
23 1+(7.70−2.06i)T+(19.9−11.5i)T2 1 + (7.70 - 2.06i)T + (19.9 - 11.5i)T^{2}
29 1+(−3.96−2.28i)T+(14.5+25.1i)T2 1 + (-3.96 - 2.28i)T + (14.5 + 25.1i)T^{2}
31 1+(3.87+3.87i)T+31iT2 1 + (3.87 + 3.87i)T + 31iT^{2}
37 1+(3.50−6.07i)T+(−18.5−32.0i)T2 1 + (3.50 - 6.07i)T + (-18.5 - 32.0i)T^{2}
41 1+(6.20−1.66i)T+(35.5−20.5i)T2 1 + (6.20 - 1.66i)T + (35.5 - 20.5i)T^{2}
43 1+(−1.67+6.24i)T+(−37.2−21.5i)T2 1 + (-1.67 + 6.24i)T + (-37.2 - 21.5i)T^{2}
47 1−0.512T+47T2 1 - 0.512T + 47T^{2}
53 1+(1.32−1.32i)T−53iT2 1 + (1.32 - 1.32i)T - 53iT^{2}
59 1+(−0.679+2.53i)T+(−51.0−29.5i)T2 1 + (-0.679 + 2.53i)T + (-51.0 - 29.5i)T^{2}
61 1+(−0.641−1.11i)T+(−30.5+52.8i)T2 1 + (-0.641 - 1.11i)T + (-30.5 + 52.8i)T^{2}
67 1+(−3.13−1.80i)T+(33.5+58.0i)T2 1 + (-3.13 - 1.80i)T + (33.5 + 58.0i)T^{2}
71 1+(1.66−6.20i)T+(−61.4−35.5i)T2 1 + (1.66 - 6.20i)T + (-61.4 - 35.5i)T^{2}
73 1−9.93iT−73T2 1 - 9.93iT - 73T^{2}
79 1+8.37iT−79T2 1 + 8.37iT - 79T^{2}
83 1+3.17T+83T2 1 + 3.17T + 83T^{2}
89 1+(−6.01+1.61i)T+(77.0−44.5i)T2 1 + (-6.01 + 1.61i)T + (77.0 - 44.5i)T^{2}
97 1+(−10.1+5.88i)T+(48.5−84.0i)T2 1 + (-10.1 + 5.88i)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

−9.949479471445969785366737453318, −8.832059385113340941147322292377, −8.038321928763751874030440521022, −7.15039414038876712199350579301, −6.20066916540710044304723409986, −5.24594346265161571205562153651, −4.08627462609593819639771872080, −3.49749773389497222016009400651, −1.31197314872825566391898586363, −0.29177821090063338201057890855, 2.32042357944865347761184731406, 3.45821622537722515980828979498, 4.32898786399189513349400169895, 5.16172626443092777875982525697, 6.61113012251247255659261611914, 7.51113693735981211105835855208, 7.969339384521973686021726227936, 9.000665252080827029168173470477, 9.831541859579686988260860740361, 10.50287249065268210449087569676

Graph of the ZZ-function along the critical line