Properties

Label 2-1400-280.69-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.894 - 0.447i$
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 + 9-s + 1.73i·11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)18-s + (−0.866 − 1.49i)22-s + i·23-s + (−0.866 − 0.499i)28-s − 1.73i·29-s + (0.866 + 0.499i)32-s + (0.499 − 0.866i)36-s + 1.73·37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s i·7-s + 0.999i·8-s + 9-s + 1.73i·11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)18-s + (−0.866 − 1.49i)22-s + i·23-s + (−0.866 − 0.499i)28-s − 1.73i·29-s + (0.866 + 0.499i)32-s + (0.499 − 0.866i)36-s + 1.73·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7978513914\)
\(L(\frac12)\) \(\approx\) \(0.7978513914\)
\(L(1)\) 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 \( 1 - T^{2} \)
11 \( 1 - 1.73iT - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.73T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
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

−9.765474911823343634028280596155, −9.289521788842409342452276023487, −7.79537218923661037734334635009, −7.56844278812981384794367257057, −6.86453488460322438690075566461, −5.95825078169335581832940843414, −4.68097393040551552482862211262, −4.11814511370719721310756726033, −2.31106944723953951983158795480, −1.23622711660360511502588434856, 1.13550396773616017793949129346, 2.49039640507032635732273873908, 3.32281008630357227150847568278, 4.44715184985955166488361882859, 5.76213838251622997730402030452, 6.50277314686949515194335874459, 7.49138597965873962041646575268, 8.353175885508261546442276795679, 8.919871072627111089026601847400, 9.559314587622905434836722568504

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