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Properties

Label 2-65-65.7-c1-0-1
Degree 22
Conductor 6565
Sign 0.810−0.585i0.810 - 0.585i
Analytic cond. 0.5190270.519027
Root an. cond. 0.7204350.720435
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  + (−0.237 − 0.137i)2-s + (0.611 + 2.28i)3-s + (−0.962 − 1.66i)4-s + (1.45 + 1.69i)5-s + (0.168 − 0.627i)6-s + (−0.193 − 0.334i)7-s + 1.07i·8-s + (−2.23 + 1.29i)9-s + (−0.112 − 0.604i)10-s + (−1.12 − 4.21i)11-s + (3.21 − 3.21i)12-s + (−1.35 − 3.34i)13-s + 0.106i·14-s + (−2.98 + 4.35i)15-s + (−1.77 + 3.07i)16-s + (1.90 + 0.510i)17-s + ⋯
L(s)  = 1  + (−0.168 − 0.0971i)2-s + (0.353 + 1.31i)3-s + (−0.481 − 0.833i)4-s + (0.650 + 0.759i)5-s + (0.0686 − 0.256i)6-s + (−0.0729 − 0.126i)7-s + 0.381i·8-s + (−0.745 + 0.430i)9-s + (−0.0356 − 0.191i)10-s + (−0.340 − 1.27i)11-s + (0.928 − 0.928i)12-s + (−0.376 − 0.926i)13-s + 0.0283i·14-s + (−0.771 + 1.12i)15-s + (−0.444 + 0.769i)16-s + (0.462 + 0.123i)17-s + ⋯

Functional equation

Λ(s)=(65s/2 ΓC(s) L(s)=((0.810−0.585i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(65s/2 ΓC(s+1/2) L(s)=((0.810−0.585i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6565    =    5⋅135 \cdot 13
Sign: 0.810−0.585i0.810 - 0.585i
Analytic conductor: 0.5190270.519027
Root analytic conductor: 0.7204350.720435
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ65(7,⋅)\chi_{65} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 65, ( :1/2), 0.810−0.585i)(2,\ 65,\ (\ :1/2),\ 0.810 - 0.585i)

Particular Values

L(1)L(1) ≈\approx 0.839230+0.271637i0.839230 + 0.271637i
L(12)L(\frac12) ≈\approx 0.839230+0.271637i0.839230 + 0.271637i
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+(−1.45−1.69i)T 1 + (-1.45 - 1.69i)T
13 1+(1.35+3.34i)T 1 + (1.35 + 3.34i)T
good2 1+(0.237+0.137i)T+(1+1.73i)T2 1 + (0.237 + 0.137i)T + (1 + 1.73i)T^{2}
3 1+(−0.611−2.28i)T+(−2.59+1.5i)T2 1 + (-0.611 - 2.28i)T + (-2.59 + 1.5i)T^{2}
7 1+(0.193+0.334i)T+(−3.5+6.06i)T2 1 + (0.193 + 0.334i)T + (-3.5 + 6.06i)T^{2}
11 1+(1.12+4.21i)T+(−9.52+5.5i)T2 1 + (1.12 + 4.21i)T + (-9.52 + 5.5i)T^{2}
17 1+(−1.90−0.510i)T+(14.7+8.5i)T2 1 + (-1.90 - 0.510i)T + (14.7 + 8.5i)T^{2}
19 1+(4.83+1.29i)T+(16.4+9.5i)T2 1 + (4.83 + 1.29i)T + (16.4 + 9.5i)T^{2}
23 1+(−0.322+0.0863i)T+(19.9−11.5i)T2 1 + (-0.322 + 0.0863i)T + (19.9 - 11.5i)T^{2}
29 1+(−7.07−4.08i)T+(14.5+25.1i)T2 1 + (-7.07 - 4.08i)T + (14.5 + 25.1i)T^{2}
31 1+(2.54+2.54i)T+31iT2 1 + (2.54 + 2.54i)T + 31iT^{2}
37 1+(2.41−4.17i)T+(−18.5−32.0i)T2 1 + (2.41 - 4.17i)T + (-18.5 - 32.0i)T^{2}
41 1+(−4.49+1.20i)T+(35.5−20.5i)T2 1 + (-4.49 + 1.20i)T + (35.5 - 20.5i)T^{2}
43 1+(1.76−6.58i)T+(−37.2−21.5i)T2 1 + (1.76 - 6.58i)T + (-37.2 - 21.5i)T^{2}
47 1+9.83T+47T2 1 + 9.83T + 47T^{2}
53 1+(7.17−7.17i)T−53iT2 1 + (7.17 - 7.17i)T - 53iT^{2}
59 1+(0.628−2.34i)T+(−51.0−29.5i)T2 1 + (0.628 - 2.34i)T + (-51.0 - 29.5i)T^{2}
61 1+(5.32+9.22i)T+(−30.5+52.8i)T2 1 + (5.32 + 9.22i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.52−3.18i)T+(33.5+58.0i)T2 1 + (-5.52 - 3.18i)T + (33.5 + 58.0i)T^{2}
71 1+(1.12−4.20i)T+(−61.4−35.5i)T2 1 + (1.12 - 4.20i)T + (-61.4 - 35.5i)T^{2}
73 1+6.08iT−73T2 1 + 6.08iT - 73T^{2}
79 1+3.34iT−79T2 1 + 3.34iT - 79T^{2}
83 1−5.18T+83T2 1 - 5.18T + 83T^{2}
89 1+(−4.82+1.29i)T+(77.0−44.5i)T2 1 + (-4.82 + 1.29i)T + (77.0 - 44.5i)T^{2}
97 1+(−12.7+7.37i)T+(48.5−84.0i)T2 1 + (-12.7 + 7.37i)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

−14.87694819067677098245980512601, −14.27392724874111161675521391470, −13.17426820957693504419246506626, −10.89840996118797164530860440018, −10.43775148195440679880050966763, −9.559414067395748037187279116498, −8.419664621509037730896008883344, −6.16688214634064142278114116429, −4.95349862258932626229744721669, −3.15299074418077516487611253347, 2.13492692750337816935576321883, 4.62618839766855619027305836632, 6.63522658477628275099557837842, 7.73757990537169006112394637857, 8.750637658077539008738670875183, 9.856896343543824856809586617313, 12.21268463114154310224424941939, 12.54549455108943206584285977656, 13.46339935611751821274948283290, 14.39283924735611274553777720797

Graph of the ZZ-function along the critical line