Properties

Label 2-1400-1400.909-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.684 - 0.728i$
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  + (−0.587 + 0.809i)2-s + (1.69 − 0.550i)3-s + (−0.309 − 0.951i)4-s + (−0.987 + 0.156i)5-s + (−0.550 + 1.69i)6-s + i·7-s + (0.951 + 0.309i)8-s + (1.76 − 1.27i)9-s + (0.453 − 0.891i)10-s + (−1.04 − 1.44i)12-s + (0.533 + 0.734i)13-s + (−0.809 − 0.587i)14-s + (−1.58 + 0.809i)15-s + (−0.809 + 0.587i)16-s + 2.17i·18-s + (−0.437 + 1.34i)19-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (1.69 − 0.550i)3-s + (−0.309 − 0.951i)4-s + (−0.987 + 0.156i)5-s + (−0.550 + 1.69i)6-s + i·7-s + (0.951 + 0.309i)8-s + (1.76 − 1.27i)9-s + (0.453 − 0.891i)10-s + (−1.04 − 1.44i)12-s + (0.533 + 0.734i)13-s + (−0.809 − 0.587i)14-s + (−1.58 + 0.809i)15-s + (−0.809 + 0.587i)16-s + 2.17i·18-s + (−0.437 + 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.684 - 0.728i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233589120\)
\(L(\frac12)\) \(\approx\) \(1.233589120\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (0.987 - 0.156i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.533 - 0.734i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (1.87 + 0.610i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.332171923033711081333646713454, −8.746628206175118287976397845247, −8.329974540434201912020050406153, −7.71238128341922892681809801741, −6.83771887591147546938983384412, −6.20959641004649640617118023101, −4.72070903555967700015510676176, −3.77238448088961556706744701029, −2.67121815734096461547153839704, −1.58018785390690581375111556196, 1.25997963002199621386917701032, 2.81623161667888085764247650052, 3.39690627252476317691984265270, 4.11221564691051548823149651257, 4.84292855668831258438538444586, 7.13820792968378167863682420903, 7.50581553574832835141140305079, 8.280599055089509959678175842569, 8.894955648713404683617235246664, 9.481014439731990175536634711217

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