Properties

Label 2-1400-5.4-c1-0-6
Degree $2$
Conductor $1400$
Sign $-0.894 - 0.447i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56i·3-s i·7-s − 3.56·9-s + 2.12·11-s + 2i·13-s + 2.56i·17-s + 0.561·19-s + 2.56·21-s + 5.56i·23-s − 1.43i·27-s − 7.56·29-s − 0.876·31-s + 5.43i·33-s + 11.8i·37-s − 5.12·39-s + ⋯
L(s)  = 1  + 1.47i·3-s − 0.377i·7-s − 1.18·9-s + 0.640·11-s + 0.554i·13-s + 0.621i·17-s + 0.128·19-s + 0.558·21-s + 1.15i·23-s − 0.276i·27-s − 1.40·29-s − 0.157·31-s + 0.946i·33-s + 1.94i·37-s − 0.820·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.343727604\)
\(L(\frac12)\) \(\approx\) \(1.343727604\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 2.56iT - 3T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 2.56iT - 17T^{2} \)
19 \( 1 - 0.561T + 19T^{2} \)
23 \( 1 - 5.56iT - 23T^{2} \)
29 \( 1 + 7.56T + 29T^{2} \)
31 \( 1 + 0.876T + 31T^{2} \)
37 \( 1 - 11.8iT - 37T^{2} \)
41 \( 1 + 6.56T + 41T^{2} \)
43 \( 1 - 2.43iT - 43T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 - 7.12iT - 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 + 16.1iT - 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + 6.68T + 79T^{2} \)
83 \( 1 + 13.6iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−9.944418976467930081910006114010, −9.235279674574032993740659397780, −8.599610225979331191475004391169, −7.51693522211801359190683919870, −6.56707188979173338674783248161, −5.56485505107817768026900609829, −4.73570518297217134361075641773, −3.88451374751306410474508577850, −3.34417556426098123028691884245, −1.65312673379399869202513228298, 0.54594894991251944640608802693, 1.81790461288747954917250439259, 2.70665487370864572860153316653, 3.95350283387045037418919886907, 5.31659199757292068096333569047, 6.03608714094064893431400892804, 6.93558943310981839460308802115, 7.42618400495319331025090667767, 8.346883187096622068435806675005, 8.995734075698349192308295999326

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