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Properties

Label 2-702-117.32-c1-0-1
Degree 22
Conductor 702702
Sign −0.949+0.315i-0.949 + 0.315i
Analytic cond. 5.605495.60549
Root an. cond. 2.367592.36759
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.653 + 0.175i)5-s + (−3.90 + 1.04i)7-s + (−0.707 + 0.707i)8-s + (−0.585 − 0.338i)10-s + (0.502 − 0.502i)11-s + (−2.29 + 2.77i)13-s + (−3.49 − 2.01i)14-s − 1.00·16-s + (−3.24 − 5.61i)17-s + (−0.253 − 0.0679i)19-s + (−0.175 − 0.653i)20-s + 0.710·22-s + (−0.860 − 1.49i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.292 + 0.0782i)5-s + (−1.47 + 0.394i)7-s + (−0.250 + 0.250i)8-s + (−0.185 − 0.106i)10-s + (0.151 − 0.151i)11-s + (−0.637 + 0.770i)13-s + (−0.934 − 0.539i)14-s − 0.250·16-s + (−0.786 − 1.36i)17-s + (−0.0581 − 0.0155i)19-s + (−0.0391 − 0.146i)20-s + 0.151·22-s + (−0.179 − 0.310i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((−0.949+0.315i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((−0.949+0.315i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −0.949+0.315i-0.949 + 0.315i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(71,⋅)\chi_{702} (71, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), −0.949+0.315i)(2,\ 702,\ (\ :1/2),\ -0.949 + 0.315i)

Particular Values

L(1)L(1) ≈\approx 0.0716758−0.443481i0.0716758 - 0.443481i
L(12)L(\frac12) ≈\approx 0.0716758−0.443481i0.0716758 - 0.443481i
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)
bad2 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
3 1 1
13 1+(2.29−2.77i)T 1 + (2.29 - 2.77i)T
good5 1+(0.653−0.175i)T+(4.33−2.5i)T2 1 + (0.653 - 0.175i)T + (4.33 - 2.5i)T^{2}
7 1+(3.90−1.04i)T+(6.06−3.5i)T2 1 + (3.90 - 1.04i)T + (6.06 - 3.5i)T^{2}
11 1+(−0.502+0.502i)T−11iT2 1 + (-0.502 + 0.502i)T - 11iT^{2}
17 1+(3.24+5.61i)T+(−8.5+14.7i)T2 1 + (3.24 + 5.61i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.253+0.0679i)T+(16.4+9.5i)T2 1 + (0.253 + 0.0679i)T + (16.4 + 9.5i)T^{2}
23 1+(0.860+1.49i)T+(−11.5+19.9i)T2 1 + (0.860 + 1.49i)T + (-11.5 + 19.9i)T^{2}
29 1+1.28iT−29T2 1 + 1.28iT - 29T^{2}
31 1+(−1.04−3.89i)T+(−26.8+15.5i)T2 1 + (-1.04 - 3.89i)T + (-26.8 + 15.5i)T^{2}
37 1+(7.96−2.13i)T+(32.0−18.5i)T2 1 + (7.96 - 2.13i)T + (32.0 - 18.5i)T^{2}
41 1+(2.39−8.94i)T+(−35.5−20.5i)T2 1 + (2.39 - 8.94i)T + (-35.5 - 20.5i)T^{2}
43 1+(5.67+3.27i)T+(21.5+37.2i)T2 1 + (5.67 + 3.27i)T + (21.5 + 37.2i)T^{2}
47 1+(−9.07−2.43i)T+(40.7+23.5i)T2 1 + (-9.07 - 2.43i)T + (40.7 + 23.5i)T^{2}
53 1−6.34iT−53T2 1 - 6.34iT - 53T^{2}
59 1+(−3.52+3.52i)T−59iT2 1 + (-3.52 + 3.52i)T - 59iT^{2}
61 1+(1.64−2.85i)T+(−30.5−52.8i)T2 1 + (1.64 - 2.85i)T + (-30.5 - 52.8i)T^{2}
67 1+(−10.0−2.70i)T+(58.0+33.5i)T2 1 + (-10.0 - 2.70i)T + (58.0 + 33.5i)T^{2}
71 1+(−2.09+7.82i)T+(−61.4−35.5i)T2 1 + (-2.09 + 7.82i)T + (-61.4 - 35.5i)T^{2}
73 1+(1.40+1.40i)T+73iT2 1 + (1.40 + 1.40i)T + 73iT^{2}
79 1+(−7.29−12.6i)T+(−39.5+68.4i)T2 1 + (-7.29 - 12.6i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.04−3.90i)T+(−71.8−41.5i)T2 1 + (1.04 - 3.90i)T + (-71.8 - 41.5i)T^{2}
89 1+(−2.64−9.85i)T+(−77.0+44.5i)T2 1 + (-2.64 - 9.85i)T + (-77.0 + 44.5i)T^{2}
97 1+(−0.520−1.94i)T+(−84.0+48.5i)T2 1 + (-0.520 - 1.94i)T + (-84.0 + 48.5i)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

−11.06981799055288198368250113273, −9.769735452453013455138216114103, −9.282597885666945050344971699306, −8.295582498845712410244352904994, −6.96412470098666158788830334339, −6.75167240978972888074488785217, −5.59920681469658752994361731358, −4.55888185989330658746537609765, −3.46948813339658232397200899562, −2.50314576324753801814143923249, 0.18331535643574576747243577127, 2.17324723788749272587626963167, 3.47706439820134354859951694677, 4.07362657029656264677000338667, 5.42190270048500667342224396990, 6.34700166799804790716485517585, 7.14364343884630910501944905512, 8.306596455324476678014167549676, 9.356121850847556945546022897042, 10.18926817635784965221583357042

Graph of the ZZ-function along the critical line