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Properties

Label 2-702-39.8-c1-0-2
Degree 22
Conductor 702702
Sign −0.819−0.572i-0.819 - 0.572i
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 + (0.730 − 0.730i)5-s + (−2.49 + 2.49i)7-s + (0.707 + 0.707i)8-s + 1.03i·10-s + (1.95 + 1.95i)11-s + (−3.21 + 1.63i)13-s − 3.53i·14-s − 1.00·16-s + 5.44·17-s + (−5.99 − 5.99i)19-s + (−0.730 − 0.730i)20-s − 2.76·22-s − 5.11·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.326 − 0.326i)5-s + (−0.943 + 0.943i)7-s + (0.250 + 0.250i)8-s + 0.326i·10-s + (0.589 + 0.589i)11-s + (−0.891 + 0.452i)13-s − 0.943i·14-s − 0.250·16-s + 1.31·17-s + (−1.37 − 1.37i)19-s + (−0.163 − 0.163i)20-s − 0.589·22-s − 1.06·23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((−0.819−0.572i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((−0.819−0.572i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −0.819−0.572i-0.819 - 0.572i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(593,⋅)\chi_{702} (593, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), −0.819−0.572i)(2,\ 702,\ (\ :1/2),\ -0.819 - 0.572i)

Particular Values

L(1)L(1) ≈\approx 0.201530+0.640605i0.201530 + 0.640605i
L(12)L(\frac12) ≈\approx 0.201530+0.640605i0.201530 + 0.640605i
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+(3.21−1.63i)T 1 + (3.21 - 1.63i)T
good5 1+(−0.730+0.730i)T−5iT2 1 + (-0.730 + 0.730i)T - 5iT^{2}
7 1+(2.49−2.49i)T−7iT2 1 + (2.49 - 2.49i)T - 7iT^{2}
11 1+(−1.95−1.95i)T+11iT2 1 + (-1.95 - 1.95i)T + 11iT^{2}
17 1−5.44T+17T2 1 - 5.44T + 17T^{2}
19 1+(5.99+5.99i)T+19iT2 1 + (5.99 + 5.99i)T + 19iT^{2}
23 1+5.11T+23T2 1 + 5.11T + 23T^{2}
29 1−8.51iT−29T2 1 - 8.51iT - 29T^{2}
31 1+(−3.53−3.53i)T+31iT2 1 + (-3.53 - 3.53i)T + 31iT^{2}
37 1+(7.78−7.78i)T−37iT2 1 + (7.78 - 7.78i)T - 37iT^{2}
41 1+(−1.06+1.06i)T−41iT2 1 + (-1.06 + 1.06i)T - 41iT^{2}
43 1−3.79iT−43T2 1 - 3.79iT - 43T^{2}
47 1+(7.80+7.80i)T+47iT2 1 + (7.80 + 7.80i)T + 47iT^{2}
53 1−11.7iT−53T2 1 - 11.7iT - 53T^{2}
59 1+(1.86+1.86i)T+59iT2 1 + (1.86 + 1.86i)T + 59iT^{2}
61 1−6.42T+61T2 1 - 6.42T + 61T^{2}
67 1+(−0.887−0.887i)T+67iT2 1 + (-0.887 - 0.887i)T + 67iT^{2}
71 1+(3.17−3.17i)T−71iT2 1 + (3.17 - 3.17i)T - 71iT^{2}
73 1+(2.31−2.31i)T−73iT2 1 + (2.31 - 2.31i)T - 73iT^{2}
79 1+5.29T+79T2 1 + 5.29T + 79T^{2}
83 1+(−4.77+4.77i)T−83iT2 1 + (-4.77 + 4.77i)T - 83iT^{2}
89 1+(−0.144−0.144i)T+89iT2 1 + (-0.144 - 0.144i)T + 89iT^{2}
97 1+(4.66+4.66i)T+97iT2 1 + (4.66 + 4.66i)T + 97iT^{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

−10.41567300298297046939205318506, −9.722286699747034430433611657456, −9.099183224406339934862309538993, −8.427440127121715629825102643984, −7.09753538487408095097912358144, −6.55806460208659242353415488798, −5.52483657531177137560321471219, −4.65697170805242917012034971939, −3.05929711693807899491072444414, −1.74328055215103137468226215348, 0.39940818260940694118374727059, 2.12750052903670536579336871964, 3.43130773221357005137772229426, 4.13637090068740132769861066329, 5.87247869420813903894750911229, 6.53825639819973943256680686388, 7.68394003319765276328635622690, 8.301645827922021875983412019173, 9.691713911421878759454607216076, 10.05439110920678122207159774985

Graph of the ZZ-function along the critical line