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Properties

Label 2-700-28.19-c1-0-11
Degree 22
Conductor 700700
Sign −0.614+0.788i-0.614 + 0.788i
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  + (−0.569 + 1.29i)2-s + (1.51 + 2.62i)3-s + (−1.35 − 1.47i)4-s + (−4.25 + 0.465i)6-s + (−2.57 − 0.602i)7-s + (2.67 − 0.908i)8-s + (−3.08 + 5.33i)9-s + (−1.03 + 0.598i)11-s + (1.82 − 5.77i)12-s + 4.83i·13-s + (2.24 − 2.99i)14-s + (−0.349 + 3.98i)16-s + (−2.20 + 1.27i)17-s + (−5.15 − 7.02i)18-s + (0.711 − 1.23i)19-s + ⋯
L(s)  = 1  + (−0.402 + 0.915i)2-s + (0.873 + 1.51i)3-s + (−0.675 − 0.737i)4-s + (−1.73 + 0.190i)6-s + (−0.973 − 0.227i)7-s + (0.946 − 0.321i)8-s + (−1.02 + 1.77i)9-s + (−0.312 + 0.180i)11-s + (0.525 − 1.66i)12-s + 1.34i·13-s + (0.600 − 0.799i)14-s + (−0.0873 + 0.996i)16-s + (−0.534 + 0.308i)17-s + (−1.21 − 1.65i)18-s + (0.163 − 0.282i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.614+0.788i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((−0.614+0.788i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.614+0.788i-0.614 + 0.788i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(551,⋅)\chi_{700} (551, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), −0.614+0.788i)(2,\ 700,\ (\ :1/2),\ -0.614 + 0.788i)

Particular Values

L(1)L(1) ≈\approx 0.382022−0.782203i0.382022 - 0.782203i
L(12)L(\frac12) ≈\approx 0.382022−0.782203i0.382022 - 0.782203i
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+(0.569−1.29i)T 1 + (0.569 - 1.29i)T
5 1 1
7 1+(2.57+0.602i)T 1 + (2.57 + 0.602i)T
good3 1+(−1.51−2.62i)T+(−1.5+2.59i)T2 1 + (-1.51 - 2.62i)T + (-1.5 + 2.59i)T^{2}
11 1+(1.03−0.598i)T+(5.5−9.52i)T2 1 + (1.03 - 0.598i)T + (5.5 - 9.52i)T^{2}
13 1−4.83iT−13T2 1 - 4.83iT - 13T^{2}
17 1+(2.20−1.27i)T+(8.5−14.7i)T2 1 + (2.20 - 1.27i)T + (8.5 - 14.7i)T^{2}
19 1+(−0.711+1.23i)T+(−9.5−16.4i)T2 1 + (-0.711 + 1.23i)T + (-9.5 - 16.4i)T^{2}
23 1+(5.02+2.90i)T+(11.5+19.9i)T2 1 + (5.02 + 2.90i)T + (11.5 + 19.9i)T^{2}
29 1−0.774T+29T2 1 - 0.774T + 29T^{2}
31 1+(3.31+5.74i)T+(−15.5+26.8i)T2 1 + (3.31 + 5.74i)T + (-15.5 + 26.8i)T^{2}
37 1+(−2.55+4.42i)T+(−18.5−32.0i)T2 1 + (-2.55 + 4.42i)T + (-18.5 - 32.0i)T^{2}
41 1−7.46iT−41T2 1 - 7.46iT - 41T^{2}
43 1−1.38iT−43T2 1 - 1.38iT - 43T^{2}
47 1+(−0.535+0.927i)T+(−23.5−40.7i)T2 1 + (-0.535 + 0.927i)T + (-23.5 - 40.7i)T^{2}
53 1+(−1.68−2.91i)T+(−26.5+45.8i)T2 1 + (-1.68 - 2.91i)T + (-26.5 + 45.8i)T^{2}
59 1+(−4.94−8.55i)T+(−29.5+51.0i)T2 1 + (-4.94 - 8.55i)T + (-29.5 + 51.0i)T^{2}
61 1+(−8.31−4.79i)T+(30.5+52.8i)T2 1 + (-8.31 - 4.79i)T + (30.5 + 52.8i)T^{2}
67 1+(9.14−5.27i)T+(33.5−58.0i)T2 1 + (9.14 - 5.27i)T + (33.5 - 58.0i)T^{2}
71 1−16.3iT−71T2 1 - 16.3iT - 71T^{2}
73 1+(0.0927−0.0535i)T+(36.5−63.2i)T2 1 + (0.0927 - 0.0535i)T + (36.5 - 63.2i)T^{2}
79 1+(9.32+5.38i)T+(39.5+68.4i)T2 1 + (9.32 + 5.38i)T + (39.5 + 68.4i)T^{2}
83 1−15.8T+83T2 1 - 15.8T + 83T^{2}
89 1+(3.41+1.97i)T+(44.5+77.0i)T2 1 + (3.41 + 1.97i)T + (44.5 + 77.0i)T^{2}
97 1−8.71iT−97T2 1 - 8.71iT - 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

−10.43265936340075645612917318602, −9.909710151926816804882544908888, −9.221198811517148732375754016325, −8.703700990271119237390897707321, −7.67662456641469550173178153635, −6.65166719918301299436218648955, −5.65439689843564083684802599268, −4.34896709510052637056794682111, −4.01758112990438524699708740074, −2.45218836886539329658567164860, 0.44935652578146817318926862882, 1.92434191703245735333548079448, 2.91825385688003622340033198172, 3.56836554748383799996661404245, 5.46793762563879348059297819629, 6.63329938220580356415099973208, 7.57164380458885044289857793900, 8.198752302041021999809221402427, 8.960899844930509936014989385568, 9.805834504073095492155394597772

Graph of the ZZ-function along the critical line