Properties

Label 2-1400-56.13-c0-0-5
Degree $2$
Conductor $1400$
Sign $1$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·6-s + 7-s − 8-s + 1.00·9-s + 1.41·12-s − 1.41·13-s − 14-s + 16-s − 1.00·18-s + 1.41·19-s + 1.41·21-s − 1.41·24-s + 1.41·26-s + 28-s − 32-s + 1.00·36-s − 1.41·38-s − 2.00·39-s − 1.41·42-s + 1.41·48-s + 49-s − 1.41·52-s − 56-s + 2.00·57-s + ⋯
L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·6-s + 7-s − 8-s + 1.00·9-s + 1.41·12-s − 1.41·13-s − 14-s + 16-s − 1.00·18-s + 1.41·19-s + 1.41·21-s − 1.41·24-s + 1.41·26-s + 28-s − 32-s + 1.00·36-s − 1.41·38-s − 2.00·39-s − 1.41·42-s + 1.41·48-s + 49-s − 1.41·52-s − 56-s + 2.00·57-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: no
Arithmetic: yes
Character: $\chi_{1400} (1301, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207780910\)
\(L(\frac12)\) \(\approx\) \(1.207780910\)
\(L(1)\) 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 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
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

−9.539879422217702921626735452615, −8.978059875023467949933676766683, −8.140939753357669301451261887400, −7.60278033682101438168505721462, −7.11327451894400425567814646014, −5.67631954318862370430598278022, −4.61969348407239127380924462246, −3.26516048459389391352190780139, −2.49304226006731570545255365727, −1.53885203955073937888686704228, 1.53885203955073937888686704228, 2.49304226006731570545255365727, 3.26516048459389391352190780139, 4.61969348407239127380924462246, 5.67631954318862370430598278022, 7.11327451894400425567814646014, 7.60278033682101438168505721462, 8.140939753357669301451261887400, 8.978059875023467949933676766683, 9.539879422217702921626735452615

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