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Properties

Label 2-702-39.5-c1-0-3
Degree 22
Conductor 702702
Sign 0.0865−0.996i0.0865 - 0.996i
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 + (−0.136 − 0.136i)5-s + (3.40 + 3.40i)7-s + (0.707 − 0.707i)8-s + 0.193i·10-s + (−2.95 + 2.95i)11-s + (−3.31 + 1.41i)13-s − 4.82i·14-s − 1.00·16-s + 1.26·17-s + (−2.82 + 2.82i)19-s + (0.136 − 0.136i)20-s + 4.17·22-s − 7.83·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.0610 − 0.0610i)5-s + (1.28 + 1.28i)7-s + (0.250 − 0.250i)8-s + 0.0610i·10-s + (−0.889 + 0.889i)11-s + (−0.919 + 0.393i)13-s − 1.28i·14-s − 0.250·16-s + 0.306·17-s + (−0.647 + 0.647i)19-s + (0.0305 − 0.0305i)20-s + 0.889·22-s − 1.63·23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((0.0865−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0865 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((0.0865−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0865 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: 0.0865−0.996i0.0865 - 0.996i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(161,⋅)\chi_{702} (161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), 0.0865−0.996i)(2,\ 702,\ (\ :1/2),\ 0.0865 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.651129+0.596995i0.651129 + 0.596995i
L(12)L(\frac12) ≈\approx 0.651129+0.596995i0.651129 + 0.596995i
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+(3.31−1.41i)T 1 + (3.31 - 1.41i)T
good5 1+(0.136+0.136i)T+5iT2 1 + (0.136 + 0.136i)T + 5iT^{2}
7 1+(−3.40−3.40i)T+7iT2 1 + (-3.40 - 3.40i)T + 7iT^{2}
11 1+(2.95−2.95i)T−11iT2 1 + (2.95 - 2.95i)T - 11iT^{2}
17 1−1.26T+17T2 1 - 1.26T + 17T^{2}
19 1+(2.82−2.82i)T−19iT2 1 + (2.82 - 2.82i)T - 19iT^{2}
23 1+7.83T+23T2 1 + 7.83T + 23T^{2}
29 1+2.30iT−29T2 1 + 2.30iT - 29T^{2}
31 1+(3.60−3.60i)T−31iT2 1 + (3.60 - 3.60i)T - 31iT^{2}
37 1+(−3.54−3.54i)T+37iT2 1 + (-3.54 - 3.54i)T + 37iT^{2}
41 1+(−1.65−1.65i)T+41iT2 1 + (-1.65 - 1.65i)T + 41iT^{2}
43 1−4.36iT−43T2 1 - 4.36iT - 43T^{2}
47 1+(−5.75+5.75i)T−47iT2 1 + (-5.75 + 5.75i)T - 47iT^{2}
53 1−8.16iT−53T2 1 - 8.16iT - 53T^{2}
59 1+(7.77−7.77i)T−59iT2 1 + (7.77 - 7.77i)T - 59iT^{2}
61 1−6.62T+61T2 1 - 6.62T + 61T^{2}
67 1+(−8.52+8.52i)T−67iT2 1 + (-8.52 + 8.52i)T - 67iT^{2}
71 1+(−4.58−4.58i)T+71iT2 1 + (-4.58 - 4.58i)T + 71iT^{2}
73 1+(6.49+6.49i)T+73iT2 1 + (6.49 + 6.49i)T + 73iT^{2}
79 1−2.03T+79T2 1 - 2.03T + 79T^{2}
83 1+(−12.1−12.1i)T+83iT2 1 + (-12.1 - 12.1i)T + 83iT^{2}
89 1+(−12.0+12.0i)T−89iT2 1 + (-12.0 + 12.0i)T - 89iT^{2}
97 1+(−5.85+5.85i)T−97iT2 1 + (-5.85 + 5.85i)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

−10.52510477207762428315992276112, −9.879262074246001905570303997842, −8.929421927709814133355046595178, −8.035914691902197255614400770802, −7.64863074415134189975423659414, −6.13961389551224442335088784435, −5.06891972856050424018566583818, −4.28251315618433470121400037813, −2.47597111379831889494074263759, −1.92703695089540771794477318040, 0.52344554303230457875350026857, 2.13581646189583132753234772835, 3.82624069443215798158961700060, 4.92178225083234635972753996545, 5.70514417392555110506857507186, 7.03324318100897431423267842319, 7.79725302271796327732887014378, 8.141045659726203014427251333295, 9.355250798267242316984577781445, 10.38904349383196503473205229001

Graph of the ZZ-function along the critical line