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Properties

Label 2-1400-1400.1021-c0-0-3
Degree $2$
Conductor $1400$
Sign $0.535 + 0.844i$
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  + (0.309 − 0.951i)2-s + (1.30 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (1.30 − 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (−0.809 − 0.587i)10-s + (−0.499 − 1.53i)12-s + (−0.5 − 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (−1.61 + 1.17i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (1.30 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (1.30 − 0.951i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.500 + 1.53i)9-s + (−0.809 − 0.587i)10-s + (−0.499 − 1.53i)12-s + (−0.5 − 1.53i)13-s + (0.309 − 0.951i)14-s + (1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + 1.61·18-s + (−1.61 + 1.17i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.898922364\)
\(L(\frac12)\) \(\approx\) \(1.898922364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 - T \)
good3 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−9.727326288159822723548727621095, −8.958785130999364203364742523217, −8.163669128052940320891796771701, −7.956953735065311140518057094329, −5.76596059249084394638027910492, −5.13968165364519432561176670455, −4.28570192734059590896901041333, −3.65088752799370993645319542647, −2.48721480605046224601032165137, −1.61389730590388008597355224516, 2.03560751753510727290825106473, 2.63002491771269250211801908806, 3.99961016097886550183992273094, 4.72790293626288624988884649695, 6.23826692587339069351911449873, 6.81778327730881742997899223889, 7.31323044258625766981827533383, 8.224991445658497893504273712044, 8.737817394874207415074434028150, 9.428714427671350971169107972370

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