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Properties

Label 2-700-35.19-c4-0-46
Degree 22
Conductor 700700
Sign −0.900−0.434i-0.900 - 0.434i
Analytic cond. 72.358972.3589
Root an. cond. 8.506408.50640
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.285 − 0.493i)3-s + (12.7 − 47.3i)7-s + (40.3 − 69.8i)9-s + (−89.1 − 154. i)11-s − 116.·13-s + (−182. − 316. i)17-s + (98.9 + 57.1i)19-s + (−26.9 + 7.21i)21-s + (−38.6 − 22.3i)23-s − 92.1·27-s − 1.12e3·29-s + (−9.18 + 5.30i)31-s + (−50.8 + 88.1i)33-s + (1.30e3 + 756. i)37-s + (33.1 + 57.3i)39-s + ⋯
L(s)  = 1  + (−0.0316 − 0.0548i)3-s + (0.259 − 0.965i)7-s + (0.497 − 0.862i)9-s + (−0.737 − 1.27i)11-s − 0.687·13-s + (−0.631 − 1.09i)17-s + (0.274 + 0.158i)19-s + (−0.0612 + 0.0163i)21-s + (−0.0730 − 0.0421i)23-s − 0.126·27-s − 1.34·29-s + (−0.00955 + 0.00551i)31-s + (−0.0467 + 0.0809i)33-s + (0.956 + 0.552i)37-s + (0.0217 + 0.0377i)39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.900−0.434i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((−0.900−0.434i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.900−0.434i-0.900 - 0.434i
Analytic conductor: 72.358972.3589
Root analytic conductor: 8.506408.50640
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ700(649,⋅)\chi_{700} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :2), −0.900−0.434i)(2,\ 700,\ (\ :2),\ -0.900 - 0.434i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.74031807520.7403180752
L(12)L(\frac12) ≈\approx 0.74031807520.7403180752
L(3)L(3) 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
5 1 1
7 1+(−12.7+47.3i)T 1 + (-12.7 + 47.3i)T
good3 1+(0.285+0.493i)T+(−40.5+70.1i)T2 1 + (0.285 + 0.493i)T + (-40.5 + 70.1i)T^{2}
11 1+(89.1+154.i)T+(−7.32e3+1.26e4i)T2 1 + (89.1 + 154. i)T + (-7.32e3 + 1.26e4i)T^{2}
13 1+116.T+2.85e4T2 1 + 116.T + 2.85e4T^{2}
17 1+(182.+316.i)T+(−4.17e4+7.23e4i)T2 1 + (182. + 316. i)T + (-4.17e4 + 7.23e4i)T^{2}
19 1+(−98.9−57.1i)T+(6.51e4+1.12e5i)T2 1 + (-98.9 - 57.1i)T + (6.51e4 + 1.12e5i)T^{2}
23 1+(38.6+22.3i)T+(1.39e5+2.42e5i)T2 1 + (38.6 + 22.3i)T + (1.39e5 + 2.42e5i)T^{2}
29 1+1.12e3T+7.07e5T2 1 + 1.12e3T + 7.07e5T^{2}
31 1+(9.18−5.30i)T+(4.61e5−7.99e5i)T2 1 + (9.18 - 5.30i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+(−1.30e3−756.i)T+(9.37e5+1.62e6i)T2 1 + (-1.30e3 - 756. i)T + (9.37e5 + 1.62e6i)T^{2}
41 1−515.iT−2.82e6T2 1 - 515. iT - 2.82e6T^{2}
43 1−2.07e3iT−3.41e6T2 1 - 2.07e3iT - 3.41e6T^{2}
47 1+(806.−1.39e3i)T+(−2.43e6−4.22e6i)T2 1 + (806. - 1.39e3i)T + (-2.43e6 - 4.22e6i)T^{2}
53 1+(−2.25e3+1.30e3i)T+(3.94e6−6.83e6i)T2 1 + (-2.25e3 + 1.30e3i)T + (3.94e6 - 6.83e6i)T^{2}
59 1+(−3.96e3+2.28e3i)T+(6.05e6−1.04e7i)T2 1 + (-3.96e3 + 2.28e3i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(−4.87e3−2.81e3i)T+(6.92e6+1.19e7i)T2 1 + (-4.87e3 - 2.81e3i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(3.19e3−1.84e3i)T+(1.00e7−1.74e7i)T2 1 + (3.19e3 - 1.84e3i)T + (1.00e7 - 1.74e7i)T^{2}
71 1+813.T+2.54e7T2 1 + 813.T + 2.54e7T^{2}
73 1+(−2.33e3−4.04e3i)T+(−1.41e7+2.45e7i)T2 1 + (-2.33e3 - 4.04e3i)T + (-1.41e7 + 2.45e7i)T^{2}
79 1+(633.−1.09e3i)T+(−1.94e7−3.37e7i)T2 1 + (633. - 1.09e3i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1+1.06e4T+4.74e7T2 1 + 1.06e4T + 4.74e7T^{2}
89 1+(8.33e3+4.81e3i)T+(3.13e7+5.43e7i)T2 1 + (8.33e3 + 4.81e3i)T + (3.13e7 + 5.43e7i)T^{2}
97 1−9.56e3T+8.85e7T2 1 - 9.56e3T + 8.85e7T^{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

−9.537614938553878979242131989830, −8.447794115424424204221421362234, −7.53064017133178218603608917140, −6.86449050897954644426697403875, −5.77854721376342499608644251390, −4.73357577675393325508745800077, −3.74336828286547567574742326469, −2.69301978798389561616227020540, −1.08694910725535467962997288193, −0.18388482165160633967992554773, 1.90472026725764601981861096549, 2.37756288013286022292427659397, 4.07434051684421468527621524692, 5.00248381227199869026504839307, 5.68188720515875183329082771957, 7.04738463370071899675206532602, 7.69762668144533430589466798759, 8.610517272901099714659248178014, 9.590188207009754299032491518615, 10.29445902573582754779733859768

Graph of the ZZ-function along the critical line