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Properties

Label 2-700-140.23-c1-0-41
Degree 22
Conductor 700700
Sign 0.902+0.430i0.902 + 0.430i
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  + (−1.20 − 0.739i)2-s + (2.47 + 0.664i)3-s + (0.906 + 1.78i)4-s + (−2.49 − 2.63i)6-s + (−1.41 − 2.23i)7-s + (0.224 − 2.81i)8-s + (3.10 + 1.79i)9-s + (1.59 − 0.921i)11-s + (1.06 + 5.02i)12-s + (2.94 + 2.94i)13-s + (0.0455 + 3.74i)14-s + (−2.35 + 3.23i)16-s + (2.96 + 0.795i)17-s + (−2.41 − 4.45i)18-s + (2.66 − 4.61i)19-s + ⋯
L(s)  = 1  + (−0.852 − 0.522i)2-s + (1.43 + 0.383i)3-s + (0.453 + 0.891i)4-s + (−1.01 − 1.07i)6-s + (−0.533 − 0.846i)7-s + (0.0793 − 0.996i)8-s + (1.03 + 0.597i)9-s + (0.481 − 0.277i)11-s + (0.307 + 1.44i)12-s + (0.817 + 0.817i)13-s + (0.0121 + 0.999i)14-s + (−0.588 + 0.808i)16-s + (0.720 + 0.192i)17-s + (−0.570 − 1.05i)18-s + (0.611 − 1.05i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.902+0.430i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((0.902+0.430i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.902+0.430i0.902 + 0.430i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(443,⋅)\chi_{700} (443, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), 0.902+0.430i)(2,\ 700,\ (\ :1/2),\ 0.902 + 0.430i)

Particular Values

L(1)L(1) ≈\approx 1.64256−0.371250i1.64256 - 0.371250i
L(12)L(\frac12) ≈\approx 1.64256−0.371250i1.64256 - 0.371250i
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+(1.20+0.739i)T 1 + (1.20 + 0.739i)T
5 1 1
7 1+(1.41+2.23i)T 1 + (1.41 + 2.23i)T
good3 1+(−2.47−0.664i)T+(2.59+1.5i)T2 1 + (-2.47 - 0.664i)T + (2.59 + 1.5i)T^{2}
11 1+(−1.59+0.921i)T+(5.5−9.52i)T2 1 + (-1.59 + 0.921i)T + (5.5 - 9.52i)T^{2}
13 1+(−2.94−2.94i)T+13iT2 1 + (-2.94 - 2.94i)T + 13iT^{2}
17 1+(−2.96−0.795i)T+(14.7+8.5i)T2 1 + (-2.96 - 0.795i)T + (14.7 + 8.5i)T^{2}
19 1+(−2.66+4.61i)T+(−9.5−16.4i)T2 1 + (-2.66 + 4.61i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.654−2.44i)T+(−19.9+11.5i)T2 1 + (-0.654 - 2.44i)T + (-19.9 + 11.5i)T^{2}
29 1+6.35iT−29T2 1 + 6.35iT - 29T^{2}
31 1+(3.78−2.18i)T+(15.5−26.8i)T2 1 + (3.78 - 2.18i)T + (15.5 - 26.8i)T^{2}
37 1+(−1.03−3.86i)T+(−32.0+18.5i)T2 1 + (-1.03 - 3.86i)T + (-32.0 + 18.5i)T^{2}
41 1−10.6T+41T2 1 - 10.6T + 41T^{2}
43 1+(−1.02+1.02i)T−43iT2 1 + (-1.02 + 1.02i)T - 43iT^{2}
47 1+(−0.785+0.210i)T+(40.7−23.5i)T2 1 + (-0.785 + 0.210i)T + (40.7 - 23.5i)T^{2}
53 1+(0.699−2.61i)T+(−45.8−26.5i)T2 1 + (0.699 - 2.61i)T + (-45.8 - 26.5i)T^{2}
59 1+(2.11+3.66i)T+(−29.5+51.0i)T2 1 + (2.11 + 3.66i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.00−10.4i)T+(−30.5−52.8i)T2 1 + (6.00 - 10.4i)T + (-30.5 - 52.8i)T^{2}
67 1+(0.856−3.19i)T+(−58.0−33.5i)T2 1 + (0.856 - 3.19i)T + (-58.0 - 33.5i)T^{2}
71 1+10.5iT−71T2 1 + 10.5iT - 71T^{2}
73 1+(−0.529+1.97i)T+(−63.2−36.5i)T2 1 + (-0.529 + 1.97i)T + (-63.2 - 36.5i)T^{2}
79 1+(3.95−6.85i)T+(−39.5−68.4i)T2 1 + (3.95 - 6.85i)T + (-39.5 - 68.4i)T^{2}
83 1+(−0.227+0.227i)T−83iT2 1 + (-0.227 + 0.227i)T - 83iT^{2}
89 1+(−3.75−2.16i)T+(44.5+77.0i)T2 1 + (-3.75 - 2.16i)T + (44.5 + 77.0i)T^{2}
97 1+(0.196−0.196i)T−97iT2 1 + (0.196 - 0.196i)T - 97iT^{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

−10.13089499702205948862147395375, −9.270481705161573494219005107138, −9.065537111655067834647912677902, −7.940549272924150827863544535684, −7.32816732495371999304862793360, −6.28008057365453456086424578973, −4.21918360073036618876404166930, −3.58576604179124881810670123026, −2.71217059043217501217093232334, −1.24621386825213704116879224174, 1.40559202643516359127549458493, 2.63442808409138624049607028261, 3.58884791124101319253014426850, 5.45214784019173952052419484204, 6.25297768165720013552613111723, 7.39804777337610795976559266180, 7.969193708803843402669861407460, 8.825223951136063560479832765652, 9.300164621658838762934604605480, 10.08529478292944289462275905817

Graph of the ZZ-function along the critical line