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Properties

Label 2-702-117.103-c1-0-0
Degree 22
Conductor 702702
Sign −0.631−0.775i-0.631 - 0.775i
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.419 − 0.242i)5-s + (−4.37 + 2.52i)7-s − 0.999i·8-s − 0.484·10-s + (−2.78 + 1.60i)11-s + (−2.69 + 2.39i)13-s + (−2.52 + 4.37i)14-s + (−0.5 − 0.866i)16-s − 4.20·17-s + 3.21i·19-s + (−0.419 + 0.242i)20-s + (−1.60 + 2.78i)22-s + (3.13 − 5.43i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.187 − 0.108i)5-s + (−1.65 + 0.955i)7-s − 0.353i·8-s − 0.153·10-s + (−0.838 + 0.484i)11-s + (−0.748 + 0.663i)13-s + (−0.675 + 1.16i)14-s + (−0.125 − 0.216i)16-s − 1.01·17-s + 0.736i·19-s + (−0.0937 + 0.0541i)20-s + (−0.342 + 0.592i)22-s + (0.653 − 1.13i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((−0.631−0.775i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((−0.631−0.775i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −0.631−0.775i-0.631 - 0.775i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(415,⋅)\chi_{702} (415, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), −0.631−0.775i)(2,\ 702,\ (\ :1/2),\ -0.631 - 0.775i)

Particular Values

L(1)L(1) ≈\approx 0.204226+0.429484i0.204226 + 0.429484i
L(12)L(\frac12) ≈\approx 0.204226+0.429484i0.204226 + 0.429484i
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1 1
13 1+(2.69−2.39i)T 1 + (2.69 - 2.39i)T
good5 1+(0.419+0.242i)T+(2.5+4.33i)T2 1 + (0.419 + 0.242i)T + (2.5 + 4.33i)T^{2}
7 1+(4.37−2.52i)T+(3.5−6.06i)T2 1 + (4.37 - 2.52i)T + (3.5 - 6.06i)T^{2}
11 1+(2.78−1.60i)T+(5.5−9.52i)T2 1 + (2.78 - 1.60i)T + (5.5 - 9.52i)T^{2}
17 1+4.20T+17T2 1 + 4.20T + 17T^{2}
19 1−3.21iT−19T2 1 - 3.21iT - 19T^{2}
23 1+(−3.13+5.43i)T+(−11.5−19.9i)T2 1 + (-3.13 + 5.43i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.29−3.97i)T+(−14.5+25.1i)T2 1 + (-2.29 - 3.97i)T + (-14.5 + 25.1i)T^{2}
31 1+(−5.61−3.24i)T+(15.5+26.8i)T2 1 + (-5.61 - 3.24i)T + (15.5 + 26.8i)T^{2}
37 1+2.08iT−37T2 1 + 2.08iT - 37T^{2}
41 1+(9.57+5.52i)T+(20.5+35.5i)T2 1 + (9.57 + 5.52i)T + (20.5 + 35.5i)T^{2}
43 1+(−4.73−8.19i)T+(−21.5+37.2i)T2 1 + (-4.73 - 8.19i)T + (-21.5 + 37.2i)T^{2}
47 1+(4.57−2.64i)T+(23.5−40.7i)T2 1 + (4.57 - 2.64i)T + (23.5 - 40.7i)T^{2}
53 1+6.41T+53T2 1 + 6.41T + 53T^{2}
59 1+(−3.13−1.81i)T+(29.5+51.0i)T2 1 + (-3.13 - 1.81i)T + (29.5 + 51.0i)T^{2}
61 1+(−0.500−0.867i)T+(−30.5+52.8i)T2 1 + (-0.500 - 0.867i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.936+0.540i)T+(33.5+58.0i)T2 1 + (0.936 + 0.540i)T + (33.5 + 58.0i)T^{2}
71 1+4.63iT−71T2 1 + 4.63iT - 71T^{2}
73 1+0.325iT−73T2 1 + 0.325iT - 73T^{2}
79 1+(−3.91−6.78i)T+(−39.5+68.4i)T2 1 + (-3.91 - 6.78i)T + (-39.5 + 68.4i)T^{2}
83 1+(5.08−2.93i)T+(41.5−71.8i)T2 1 + (5.08 - 2.93i)T + (41.5 - 71.8i)T^{2}
89 1−8.42iT−89T2 1 - 8.42iT - 89T^{2}
97 1+(11.3−6.52i)T+(48.5−84.0i)T2 1 + (11.3 - 6.52i)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

−10.67278379429178454209509857120, −9.960534285918928967329891464562, −9.234592667267716340702479494409, −8.284045664379738113592657519627, −6.83773440265483659388760886241, −6.40255903333052342627001268063, −5.22828561398951896532092644053, −4.33112163263410638532679615608, −3.01015524717142024409517368296, −2.30155635757963874050790962650, 0.18386750712169190327341470860, 2.77045732956011498307999739790, 3.45884369877892890602650899441, 4.61157382350703034750714461498, 5.67646364497401602550835552117, 6.68108029837614979575747574003, 7.25252872878618271059258547667, 8.170490247908312765425973785015, 9.434981011887080932736844173018, 10.12104699506930228942792779353

Graph of the ZZ-function along the critical line