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Properties

Label 2-700-28.3-c1-0-20
Degree 22
Conductor 700700
Sign 0.980−0.196i0.980 - 0.196i
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  + (−1.37 − 0.334i)2-s + (0.556 − 0.963i)3-s + (1.77 + 0.919i)4-s + (−1.08 + 1.13i)6-s + (2.32 + 1.26i)7-s + (−2.13 − 1.85i)8-s + (0.880 + 1.52i)9-s + (−1.48 − 0.856i)11-s + (1.87 − 1.20i)12-s + 2.45i·13-s + (−2.77 − 2.51i)14-s + (2.30 + 3.26i)16-s + (5.38 + 3.10i)17-s + (−0.699 − 2.39i)18-s + (0.108 + 0.187i)19-s + ⋯
L(s)  = 1  + (−0.971 − 0.236i)2-s + (0.321 − 0.556i)3-s + (0.888 + 0.459i)4-s + (−0.443 + 0.464i)6-s + (0.878 + 0.477i)7-s + (−0.753 − 0.656i)8-s + (0.293 + 0.508i)9-s + (−0.447 − 0.258i)11-s + (0.541 − 0.346i)12-s + 0.682i·13-s + (−0.740 − 0.672i)14-s + (0.577 + 0.816i)16-s + (1.30 + 0.754i)17-s + (−0.164 − 0.563i)18-s + (0.0248 + 0.0430i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.980−0.196i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((0.980−0.196i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.980−0.196i0.980 - 0.196i
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), 0.980−0.196i)(2,\ 700,\ (\ :1/2),\ 0.980 - 0.196i)

Particular Values

L(1)L(1) ≈\approx 1.19747+0.118900i1.19747 + 0.118900i
L(12)L(\frac12) ≈\approx 1.19747+0.118900i1.19747 + 0.118900i
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+(1.37+0.334i)T 1 + (1.37 + 0.334i)T
5 1 1
7 1+(−2.32−1.26i)T 1 + (-2.32 - 1.26i)T
good3 1+(−0.556+0.963i)T+(−1.5−2.59i)T2 1 + (-0.556 + 0.963i)T + (-1.5 - 2.59i)T^{2}
11 1+(1.48+0.856i)T+(5.5+9.52i)T2 1 + (1.48 + 0.856i)T + (5.5 + 9.52i)T^{2}
13 1−2.45iT−13T2 1 - 2.45iT - 13T^{2}
17 1+(−5.38−3.10i)T+(8.5+14.7i)T2 1 + (-5.38 - 3.10i)T + (8.5 + 14.7i)T^{2}
19 1+(−0.108−0.187i)T+(−9.5+16.4i)T2 1 + (-0.108 - 0.187i)T + (-9.5 + 16.4i)T^{2}
23 1+(5.68−3.28i)T+(11.5−19.9i)T2 1 + (5.68 - 3.28i)T + (11.5 - 19.9i)T^{2}
29 1+2.47T+29T2 1 + 2.47T + 29T^{2}
31 1+(0.0819−0.141i)T+(−15.5−26.8i)T2 1 + (0.0819 - 0.141i)T + (-15.5 - 26.8i)T^{2}
37 1+(−3.84−6.66i)T+(−18.5+32.0i)T2 1 + (-3.84 - 6.66i)T + (-18.5 + 32.0i)T^{2}
41 1−8.34iT−41T2 1 - 8.34iT - 41T^{2}
43 1+1.89iT−43T2 1 + 1.89iT - 43T^{2}
47 1+(5.85+10.1i)T+(−23.5+40.7i)T2 1 + (5.85 + 10.1i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6.51+11.2i)T+(−26.5−45.8i)T2 1 + (-6.51 + 11.2i)T + (-26.5 - 45.8i)T^{2}
59 1+(−2.14+3.71i)T+(−29.5−51.0i)T2 1 + (-2.14 + 3.71i)T + (-29.5 - 51.0i)T^{2}
61 1+(−6.06+3.50i)T+(30.5−52.8i)T2 1 + (-6.06 + 3.50i)T + (30.5 - 52.8i)T^{2}
67 1+(−4.48−2.58i)T+(33.5+58.0i)T2 1 + (-4.48 - 2.58i)T + (33.5 + 58.0i)T^{2}
71 1−5.04iT−71T2 1 - 5.04iT - 71T^{2}
73 1+(6.59+3.80i)T+(36.5+63.2i)T2 1 + (6.59 + 3.80i)T + (36.5 + 63.2i)T^{2}
79 1+(−13.8+7.97i)T+(39.5−68.4i)T2 1 + (-13.8 + 7.97i)T + (39.5 - 68.4i)T^{2}
83 1−5.47T+83T2 1 - 5.47T + 83T^{2}
89 1+(1.54−0.891i)T+(44.5−77.0i)T2 1 + (1.54 - 0.891i)T + (44.5 - 77.0i)T^{2}
97 1−10.5iT−97T2 1 - 10.5iT - 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

−10.29133191569077621708080718438, −9.711871548587242080521328078013, −8.383620175470348688270859309883, −8.141773469636662859659514799852, −7.35839119720958587056653737846, −6.25783434747594788602343040883, −5.15976875856609973765509964886, −3.62510748193804276405241577353, −2.24418774841745207718404980376, −1.47510166339658242695316087543, 0.928283200296125673111796119193, 2.49750166112876960508207664648, 3.81826333670836245822813285126, 5.05990115044188546331080425793, 6.02657376685861646229826443249, 7.36772969455376032605663153334, 7.78331976300229446534430103802, 8.732803172412292727623581671524, 9.643883237181248029363792254060, 10.24148316505262852525374062213

Graph of the ZZ-function along the critical line