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Properties

Label 2-700-28.19-c1-0-39
Degree 22
Conductor 700700
Sign 0.506−0.862i0.506 - 0.862i
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  + (−0.288 + 1.38i)2-s + (0.450 + 0.780i)3-s + (−1.83 − 0.798i)4-s + (−1.21 + 0.398i)6-s + (2.29 + 1.30i)7-s + (1.63 − 2.30i)8-s + (1.09 − 1.89i)9-s + (3.24 − 1.87i)11-s + (−0.202 − 1.79i)12-s − 2.41i·13-s + (−2.47 + 2.80i)14-s + (2.72 + 2.92i)16-s + (0.505 − 0.291i)17-s + (2.30 + 2.06i)18-s + (3.07 − 5.33i)19-s + ⋯
L(s)  = 1  + (−0.204 + 0.978i)2-s + (0.260 + 0.450i)3-s + (−0.916 − 0.399i)4-s + (−0.494 + 0.162i)6-s + (0.869 + 0.494i)7-s + (0.578 − 0.815i)8-s + (0.364 − 0.631i)9-s + (0.977 − 0.564i)11-s + (−0.0585 − 0.517i)12-s − 0.671i·13-s + (−0.661 + 0.750i)14-s + (0.680 + 0.732i)16-s + (0.122 − 0.0707i)17-s + (0.543 + 0.485i)18-s + (0.706 − 1.22i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.506−0.862i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((0.506−0.862i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.506−0.862i0.506 - 0.862i
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), 0.506−0.862i)(2,\ 700,\ (\ :1/2),\ 0.506 - 0.862i)

Particular Values

L(1)L(1) ≈\approx 1.40383+0.803268i1.40383 + 0.803268i
L(12)L(\frac12) ≈\approx 1.40383+0.803268i1.40383 + 0.803268i
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+(0.288−1.38i)T 1 + (0.288 - 1.38i)T
5 1 1
7 1+(−2.29−1.30i)T 1 + (-2.29 - 1.30i)T
good3 1+(−0.450−0.780i)T+(−1.5+2.59i)T2 1 + (-0.450 - 0.780i)T + (-1.5 + 2.59i)T^{2}
11 1+(−3.24+1.87i)T+(5.5−9.52i)T2 1 + (-3.24 + 1.87i)T + (5.5 - 9.52i)T^{2}
13 1+2.41iT−13T2 1 + 2.41iT - 13T^{2}
17 1+(−0.505+0.291i)T+(8.5−14.7i)T2 1 + (-0.505 + 0.291i)T + (8.5 - 14.7i)T^{2}
19 1+(−3.07+5.33i)T+(−9.5−16.4i)T2 1 + (-3.07 + 5.33i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.73+2.15i)T+(11.5+19.9i)T2 1 + (3.73 + 2.15i)T + (11.5 + 19.9i)T^{2}
29 1+0.435T+29T2 1 + 0.435T + 29T^{2}
31 1+(−1.26−2.19i)T+(−15.5+26.8i)T2 1 + (-1.26 - 2.19i)T + (-15.5 + 26.8i)T^{2}
37 1+(5.65−9.78i)T+(−18.5−32.0i)T2 1 + (5.65 - 9.78i)T + (-18.5 - 32.0i)T^{2}
41 1−7.35iT−41T2 1 - 7.35iT - 41T^{2}
43 1−5.80iT−43T2 1 - 5.80iT - 43T^{2}
47 1+(−5.78+10.0i)T+(−23.5−40.7i)T2 1 + (-5.78 + 10.0i)T + (-23.5 - 40.7i)T^{2}
53 1+(1.55+2.69i)T+(−26.5+45.8i)T2 1 + (1.55 + 2.69i)T + (-26.5 + 45.8i)T^{2}
59 1+(−1.73−3.00i)T+(−29.5+51.0i)T2 1 + (-1.73 - 3.00i)T + (-29.5 + 51.0i)T^{2}
61 1+(8.99+5.19i)T+(30.5+52.8i)T2 1 + (8.99 + 5.19i)T + (30.5 + 52.8i)T^{2}
67 1+(−8.52+4.92i)T+(33.5−58.0i)T2 1 + (-8.52 + 4.92i)T + (33.5 - 58.0i)T^{2}
71 1−9.96iT−71T2 1 - 9.96iT - 71T^{2}
73 1+(8.48−4.89i)T+(36.5−63.2i)T2 1 + (8.48 - 4.89i)T + (36.5 - 63.2i)T^{2}
79 1+(−0.397−0.229i)T+(39.5+68.4i)T2 1 + (-0.397 - 0.229i)T + (39.5 + 68.4i)T^{2}
83 1+2.59T+83T2 1 + 2.59T + 83T^{2}
89 1+(8.55+4.94i)T+(44.5+77.0i)T2 1 + (8.55 + 4.94i)T + (44.5 + 77.0i)T^{2}
97 1+4.54iT−97T2 1 + 4.54iT - 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

−10.26868032315461018657650697784, −9.517320483380932721221831261726, −8.724924930497671790525700281345, −8.193359014806216574932042231883, −7.03984878064138429624432859154, −6.23258650267539193696138988422, −5.18473240947939256646899729109, −4.37439883188040736035252330151, −3.21440342933127530425380718191, −1.13199219369193030973908726704, 1.42747652266068465734193393270, 2.06334771878622122453001655337, 3.78140005486574857981845541544, 4.41969111333366381119231629663, 5.61994972933546055765557827627, 7.24505093422419116895140097668, 7.70458216584139071539264597139, 8.685078148088543470546396724485, 9.552315658691274286267401492955, 10.39295363427268366571405206908

Graph of the ZZ-function along the critical line