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Properties

Label 2-1400-56.11-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.832 + 0.553i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·7-s − 0.999i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (−0.5 − 0.866i)19-s − 0.999i·22-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)26-s + (0.866 + 0.499i)28-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + i·7-s − 0.999i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (−0.5 − 0.866i)19-s − 0.999i·22-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)26-s + (0.866 + 0.499i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.856041779\)
\(L(\frac12)\) \(\approx\) \(1.856041779\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.726579668190755514421777113304, −9.006632373163544411120614787731, −8.203925436163861706452882374116, −6.80149828697297067313718488760, −6.44410319688415009718417845928, −5.21939050070547900073886043687, −4.75170224060472153993523193261, −3.60539332497162277590541150234, −2.56916313280053855079446431611, −1.64375435082450942846861992013, 1.58105235139067307344594910014, 3.22092628046111606104817532526, 3.89944864340438923751048250794, 4.69796863583890074529305885822, 5.69772431540682343415447595872, 6.69366977624220667306758680083, 7.16131993941169077278009588275, 7.937339122557652332037629190499, 8.914721527312693272260308244954, 9.967066307115873797552771171660

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