Properties

Label 2-1400-56.13-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.295024346\)
\(L(\frac12)\) \(\approx\) \(1.295024346\)
\(L(1)\) 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

−10.10337105892282530438826706684, −9.128431405420742852408896083804, −7.80062367076982391498526349108, −6.87119116955904076246851068767, −6.20722443103545889364585251590, −5.73041708354120256504712037816, −4.89543848695057668836629153368, −3.82834728250147071235564423871, −3.00160949841360440982666450832, −1.24202895758901682203803852052, 1.24202895758901682203803852052, 3.00160949841360440982666450832, 3.82834728250147071235564423871, 4.89543848695057668836629153368, 5.73041708354120256504712037816, 6.20722443103545889364585251590, 6.87119116955904076246851068767, 7.80062367076982391498526349108, 9.128431405420742852408896083804, 10.10337105892282530438826706684

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