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Properties

Label 2-702-117.11-c1-0-11
Degree 22
Conductor 702702
Sign −0.760+0.649i-0.760 + 0.649i
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 + (−1.35 − 0.361i)5-s + (0.977 + 0.261i)7-s + (0.707 + 0.707i)8-s + (1.21 − 0.699i)10-s + (−4.11 − 4.11i)11-s + (3.22 + 1.61i)13-s + (−0.876 + 0.505i)14-s − 1.00·16-s + (−1.67 + 2.89i)17-s + (−7.07 + 1.89i)19-s + (−0.361 + 1.35i)20-s + 5.81·22-s + (−0.290 + 0.502i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.603 − 0.161i)5-s + (0.369 + 0.0989i)7-s + (0.250 + 0.250i)8-s + (0.382 − 0.221i)10-s + (−1.23 − 1.23i)11-s + (0.894 + 0.446i)13-s + (−0.234 + 0.135i)14-s − 0.250·16-s + (−0.405 + 0.702i)17-s + (−1.62 + 0.434i)19-s + (−0.0809 + 0.301i)20-s + 1.23·22-s + (−0.0605 + 0.104i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((−0.760+0.649i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((−0.760+0.649i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −0.760+0.649i-0.760 + 0.649i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(89,⋅)\chi_{702} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), −0.760+0.649i)(2,\ 702,\ (\ :1/2),\ -0.760 + 0.649i)

Particular Values

L(1)L(1) ≈\approx 0.0586624−0.158916i0.0586624 - 0.158916i
L(12)L(\frac12) ≈\approx 0.0586624−0.158916i0.0586624 - 0.158916i
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+(−3.22−1.61i)T 1 + (-3.22 - 1.61i)T
good5 1+(1.35+0.361i)T+(4.33+2.5i)T2 1 + (1.35 + 0.361i)T + (4.33 + 2.5i)T^{2}
7 1+(−0.977−0.261i)T+(6.06+3.5i)T2 1 + (-0.977 - 0.261i)T + (6.06 + 3.5i)T^{2}
11 1+(4.11+4.11i)T+11iT2 1 + (4.11 + 4.11i)T + 11iT^{2}
17 1+(1.67−2.89i)T+(−8.5−14.7i)T2 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2}
19 1+(7.07−1.89i)T+(16.4−9.5i)T2 1 + (7.07 - 1.89i)T + (16.4 - 9.5i)T^{2}
23 1+(0.290−0.502i)T+(−11.5−19.9i)T2 1 + (0.290 - 0.502i)T + (-11.5 - 19.9i)T^{2}
29 1−1.03iT−29T2 1 - 1.03iT - 29T^{2}
31 1+(−2.28+8.52i)T+(−26.8−15.5i)T2 1 + (-2.28 + 8.52i)T + (-26.8 - 15.5i)T^{2}
37 1+(7.82+2.09i)T+(32.0+18.5i)T2 1 + (7.82 + 2.09i)T + (32.0 + 18.5i)T^{2}
41 1+(−0.423−1.58i)T+(−35.5+20.5i)T2 1 + (-0.423 - 1.58i)T + (-35.5 + 20.5i)T^{2}
43 1+(7.25−4.19i)T+(21.5−37.2i)T2 1 + (7.25 - 4.19i)T + (21.5 - 37.2i)T^{2}
47 1+(−1.45+0.388i)T+(40.7−23.5i)T2 1 + (-1.45 + 0.388i)T + (40.7 - 23.5i)T^{2}
53 1−2.77iT−53T2 1 - 2.77iT - 53T^{2}
59 1+(8.47+8.47i)T+59iT2 1 + (8.47 + 8.47i)T + 59iT^{2}
61 1+(1.41+2.44i)T+(−30.5+52.8i)T2 1 + (1.41 + 2.44i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.09+0.293i)T+(58.0−33.5i)T2 1 + (-1.09 + 0.293i)T + (58.0 - 33.5i)T^{2}
71 1+(2.71+10.1i)T+(−61.4+35.5i)T2 1 + (2.71 + 10.1i)T + (-61.4 + 35.5i)T^{2}
73 1+(−0.788+0.788i)T−73iT2 1 + (-0.788 + 0.788i)T - 73iT^{2}
79 1+(0.827−1.43i)T+(−39.5−68.4i)T2 1 + (0.827 - 1.43i)T + (-39.5 - 68.4i)T^{2}
83 1+(−4.21−15.7i)T+(−71.8+41.5i)T2 1 + (-4.21 - 15.7i)T + (-71.8 + 41.5i)T^{2}
89 1+(−0.783+2.92i)T+(−77.0−44.5i)T2 1 + (-0.783 + 2.92i)T + (-77.0 - 44.5i)T^{2}
97 1+(−3.18+11.8i)T+(−84.0−48.5i)T2 1 + (-3.18 + 11.8i)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

−10.19953803739303201928875225786, −8.911227181115109832140725616140, −8.210302481758641788693051073930, −7.934141734301669158008013744009, −6.46169061293610271800679995921, −5.87158543716321264539943007075, −4.65740978246793990361391535271, −3.59713040623507165239471185989, −1.96268524714759776524458377385, −0.099267946391601268014855845269, 1.86458940744360179304135161802, 3.03404073686163148555278710306, 4.27664881366779780372709009436, 5.12356519351037258448936098399, 6.65690133120096862664653200765, 7.47189840401343502719368624132, 8.271899416912659131246575785049, 8.954648020580573897424118479614, 10.27688699098232459096634801898, 10.56804203614466154853420807522

Graph of the ZZ-function along the critical line