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Properties

Label 2-700-28.3-c1-0-33
Degree 22
Conductor 700700
Sign 0.890+0.455i0.890 + 0.455i
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  + (−0.836 − 1.14i)2-s + (−1.51 + 2.62i)3-s + (−0.601 + 1.90i)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 + (−4.08 − 4.46i)12-s − 4.83i·13-s + (−2.84 − 2.43i)14-s + (−3.27 − 2.29i)16-s + (−2.20 − 1.27i)17-s + (−3.51 + 7.97i)18-s + (−0.711 − 1.23i)19-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (−0.873 + 1.51i)3-s + (−0.300 + 0.953i)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 + (−1.18 − 1.28i)12-s − 1.34i·13-s + (−0.759 − 0.650i)14-s + (−0.819 − 0.573i)16-s + (−0.534 − 0.308i)17-s + (−0.827 + 1.88i)18-s + (−0.163 − 0.282i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.890+0.455i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((0.890+0.455i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.890+0.455i0.890 + 0.455i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(451,⋅)\chi_{700} (451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), 0.890+0.455i)(2,\ 700,\ (\ :1/2),\ 0.890 + 0.455i)

Particular Values

L(1)L(1) ≈\approx 0.792364−0.190766i0.792364 - 0.190766i
L(12)L(\frac12) ≈\approx 0.792364−0.190766i0.792364 - 0.190766i
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+(0.836+1.14i)T 1 + (0.836 + 1.14i)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

−10.40210528331998702562922061893, −9.863279266647859488813551180964, −8.904847367369188795314248519680, −8.177271795593449647223613949268, −6.94101056625700683076648265758, −5.49391049114834537016936785505, −4.68273745736973960511899879060, −3.98460892553047401986061799485, −2.72468905740558807866741412921, −0.71528957953562981778827753616, 1.17778384430547960049771104980, 2.01260622243891480547895220090, 4.56406564564490900799819343384, 5.43334586465029872316603625734, 6.41162280951057939662421517434, 6.89028373267100292074836393485, 7.77808380768181141830520709699, 8.505130390455688697616747087020, 9.350509392177133406067427743510, 10.80914163934672621410838390612

Graph of the ZZ-function along the critical line