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Properties

Label 2-700-35.19-c4-0-45
Degree $2$
Conductor $700$
Sign $0.481 - 0.876i$
Analytic cond. $72.3589$
Root an. cond. $8.50640$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.52 − 14.7i)3-s + (40.1 − 28.1i)7-s + (−104. + 181. i)9-s + (−93.3 − 161. i)11-s + 200.·13-s + (−55.0 − 95.3i)17-s + (226. + 130. i)19-s + (−757. − 352. i)21-s + (−385. − 222. i)23-s + 2.18e3·27-s − 734.·29-s + (−406. + 234. i)31-s + (−1.59e3 + 2.75e3i)33-s + (−1.74e3 − 1.00e3i)37-s + (−1.71e3 − 2.96e3i)39-s + ⋯
L(s)  = 1  + (−0.946 − 1.64i)3-s + (0.818 − 0.574i)7-s + (−1.29 + 2.23i)9-s + (−0.771 − 1.33i)11-s + 1.18·13-s + (−0.190 − 0.329i)17-s + (0.626 + 0.361i)19-s + (−1.71 − 0.799i)21-s + (−0.727 − 0.420i)23-s + 3.00·27-s − 0.873·29-s + (−0.422 + 0.244i)31-s + (−1.46 + 2.52i)33-s + (−1.27 − 0.735i)37-s + (−1.12 − 1.94i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(72.3589\)
Root analytic conductor: \(8.50640\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :2),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.03873714477\)
\(L(\frac12)\) \(\approx\) \(0.03873714477\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-40.1 + 28.1i)T \)
good3 \( 1 + (8.52 + 14.7i)T + (-40.5 + 70.1i)T^{2} \)
11 \( 1 + (93.3 + 161. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 - 200.T + 2.85e4T^{2} \)
17 \( 1 + (55.0 + 95.3i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-226. - 130. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (385. + 222. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 734.T + 7.07e5T^{2} \)
31 \( 1 + (406. - 234. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (1.74e3 + 1.00e3i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 46.3iT - 2.82e6T^{2} \)
43 \( 1 - 2.74e3iT - 3.41e6T^{2} \)
47 \( 1 + (2.14e3 - 3.71e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.85e3 - 1.64e3i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-207. + 119. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.36e3 + 1.36e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-4.32e3 + 2.49e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 2.96e3T + 2.54e7T^{2} \)
73 \( 1 + (2.08e3 + 3.60e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-1.76e3 + 3.05e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + 2.15e3T + 4.74e7T^{2} \)
89 \( 1 + (8.80e3 + 5.08e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 - 1.20e4T + 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.778658390587201294204727335781, −7.918924236319869074821856277381, −7.55026009034975205299663902481, −6.33476337397327589801005023508, −5.81267519886885297523542820227, −4.88155401814379669647413819182, −3.26837589391737576220336067425, −1.81465262854078303684712654760, −1.00315600308214764943368121749, −0.01188622227613067052854266146, 1.81957876509016754811152417419, 3.44749953374816190913432521748, 4.33937054026669557792651656144, 5.23756277113857683739319712095, 5.65450397183183544988223500394, 6.91540341942404823271912165411, 8.243021402107667867685429430687, 9.054937135511830020222735769340, 9.923974898270311712392853141870, 10.50339078694995105223684667581

Graph of the $Z$-function along the critical line