Properties

Label 2-1400-56.13-c0-0-6
Degree $2$
Conductor $1400$
Sign $1$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 9-s + 14-s + 16-s − 18-s − 2·23-s + 28-s + 32-s − 36-s − 2·46-s + 49-s + 56-s − 63-s + 64-s − 2·71-s − 72-s − 2·79-s + 81-s − 2·92-s + 98-s + 112-s + 2·113-s + ⋯
L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 9-s + 14-s + 16-s − 18-s − 2·23-s + 28-s + 32-s − 36-s − 2·46-s + 49-s + 56-s − 63-s + 64-s − 2·71-s − 72-s − 2·79-s + 81-s − 2·92-s + 98-s + 112-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1400} (1301, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.081859889\)
\(L(\frac12)\) \(\approx\) \(2.081859889\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.00882980473781395286441698400, −8.725031599692830244359459696755, −8.037251577557102400555611085397, −7.32122865117550002331936504498, −6.14555230718548433550421110794, −5.62579229132141023663837039834, −4.69192565931160717702699352737, −3.87274410987393983486408048515, −2.73043243486413692739991743345, −1.75125247108869023879522052870, 1.75125247108869023879522052870, 2.73043243486413692739991743345, 3.87274410987393983486408048515, 4.69192565931160717702699352737, 5.62579229132141023663837039834, 6.14555230718548433550421110794, 7.32122865117550002331936504498, 8.037251577557102400555611085397, 8.725031599692830244359459696755, 10.00882980473781395286441698400

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