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Properties

Label 2-700-140.59-c1-0-22
Degree 22
Conductor 700700
Sign −0.389−0.920i-0.389 - 0.920i
Analytic cond. 5.589525.58952
Root an. cond. 2.364212.36421
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  + (−0.366 + 1.36i)2-s + (1.5 + 0.866i)3-s + (−1.73 − i)4-s + (−1.73 + 1.73i)6-s + (2 + 1.73i)7-s + (2 − 1.99i)8-s + (0.866 + 0.5i)11-s + (−1.73 − 3i)12-s + 3.46·13-s + (−3.09 + 2.09i)14-s + (1.99 + 3.46i)16-s + (−0.866 + 1.5i)17-s + (2.59 + 4.5i)19-s + (1.50 + 4.33i)21-s + (−1 + 0.999i)22-s + (0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (0.866 + 0.499i)3-s + (−0.866 − 0.5i)4-s + (−0.707 + 0.707i)6-s + (0.755 + 0.654i)7-s + (0.707 − 0.707i)8-s + (0.261 + 0.150i)11-s + (−0.499 − 0.866i)12-s + 0.960·13-s + (−0.827 + 0.560i)14-s + (0.499 + 0.866i)16-s + (−0.210 + 0.363i)17-s + (0.596 + 1.03i)19-s + (0.327 + 0.944i)21-s + (−0.213 + 0.213i)22-s + (0.104 + 0.180i)23-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.389−0.920i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((−0.389−0.920i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.389−0.920i-0.389 - 0.920i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(199,⋅)\chi_{700} (199, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), −0.389−0.920i)(2,\ 700,\ (\ :1/2),\ -0.389 - 0.920i)

Particular Values

L(1)L(1) ≈\approx 0.985908+1.48768i0.985908 + 1.48768i
L(12)L(\frac12) ≈\approx 0.985908+1.48768i0.985908 + 1.48768i
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+(0.366−1.36i)T 1 + (0.366 - 1.36i)T
5 1 1
7 1+(−2−1.73i)T 1 + (-2 - 1.73i)T
good3 1+(−1.5−0.866i)T+(1.5+2.59i)T2 1 + (-1.5 - 0.866i)T + (1.5 + 2.59i)T^{2}
11 1+(−0.866−0.5i)T+(5.5+9.52i)T2 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2}
13 1−3.46T+13T2 1 - 3.46T + 13T^{2}
17 1+(0.866−1.5i)T+(−8.5−14.7i)T2 1 + (0.866 - 1.5i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.59−4.5i)T+(−9.5+16.4i)T2 1 + (-2.59 - 4.5i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.5−0.866i)T+(−11.5+19.9i)T2 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1+(0.866−1.5i)T+(−15.5−26.8i)T2 1 + (0.866 - 1.5i)T + (-15.5 - 26.8i)T^{2}
37 1+(2.59−1.5i)T+(18.5−32.0i)T2 1 + (2.59 - 1.5i)T + (18.5 - 32.0i)T^{2}
41 1−3.46iT−41T2 1 - 3.46iT - 41T^{2}
43 1+2T+43T2 1 + 2T + 43T^{2}
47 1+(−7.5+4.33i)T+(23.5−40.7i)T2 1 + (-7.5 + 4.33i)T + (23.5 - 40.7i)T^{2}
53 1+(−0.866−0.5i)T+(26.5+45.8i)T2 1 + (-0.866 - 0.5i)T + (26.5 + 45.8i)T^{2}
59 1+(−2.59+4.5i)T+(−29.5−51.0i)T2 1 + (-2.59 + 4.5i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.5−2.59i)T+(30.5−52.8i)T2 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2}
67 1+(−1.5+2.59i)T+(−33.5−58.0i)T2 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2}
71 1+14iT−71T2 1 + 14iT - 71T^{2}
73 1+(4.33−7.5i)T+(−36.5−63.2i)T2 1 + (4.33 - 7.5i)T + (-36.5 - 63.2i)T^{2}
79 1+(7.79−4.5i)T+(39.5−68.4i)T2 1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2}
83 1+13.8iT−83T2 1 + 13.8iT - 83T^{2}
89 1+(13.5−7.79i)T+(44.5−77.0i)T2 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2}
97 1−17.3T+97T2 1 - 17.3T + 97T^{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

−10.39450376470148303005216250693, −9.522606076433167199569821060406, −8.784390268790069002135213989887, −8.329238989130423596610684774755, −7.45856296287991111435719889427, −6.21675500437940924871750369188, −5.47367602337224744985920401331, −4.29793959054472569820814567109, −3.40627642382740140959326295056, −1.63615869036582801559625912839, 1.08973939949667980379347666164, 2.22036206220545976712334506146, 3.32184001676576729709230177566, 4.30404233087412887125534308268, 5.42407913160834540128763876919, 7.11173847610861189008707184930, 7.74538544110410122402442366655, 8.690398869103533851808974031846, 9.088509394964883773528698274024, 10.29810953477229359713692446140

Graph of the ZZ-function along the critical line