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Properties

Label 2-1568-8.3-c0-0-1
Degree $2$
Conductor $1568$
Sign $1$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
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  − 9-s + 2·11-s + 25-s + 2·43-s − 2·67-s + 81-s − 2·99-s − 2·107-s − 2·113-s + ⋯
L(s)  = 1  − 9-s + 2·11-s + 25-s + 2·43-s − 2·67-s + 81-s − 2·99-s − 2·107-s − 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1568} (687, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.162224734\)
\(L(\frac12)\) \(\approx\) \(1.162224734\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.227635430459730001050738630240, −9.093956396849252373149756368451, −8.146519254255312905606049695108, −7.11575839914160251909644175925, −6.37000837951730031424901005124, −5.67404974498474098311751262066, −4.52821896107356616481889748549, −3.68102708316251376029199155538, −2.65822881295695467178310367079, −1.24937716619636972835746446093, 1.24937716619636972835746446093, 2.65822881295695467178310367079, 3.68102708316251376029199155538, 4.52821896107356616481889748549, 5.67404974498474098311751262066, 6.37000837951730031424901005124, 7.11575839914160251909644175925, 8.146519254255312905606049695108, 9.093956396849252373149756368451, 9.227635430459730001050738630240

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