Properties

Label 2-1568-8.3-c2-0-26
Degree $2$
Conductor $1568$
Sign $1$
Analytic cond. $42.7249$
Root an. cond. $6.53642$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5·9-s − 14·11-s − 2·17-s − 34·19-s + 25·25-s + 28·27-s + 28·33-s + 46·41-s − 14·43-s + 4·51-s + 68·57-s − 82·59-s − 62·67-s + 142·73-s − 50·75-s − 11·81-s + 158·83-s − 146·89-s + 94·97-s + 70·99-s + 178·107-s + 98·113-s + ⋯
L(s)  = 1  − 2/3·3-s − 5/9·9-s − 1.27·11-s − 0.117·17-s − 1.78·19-s + 25-s + 1.03·27-s + 0.848·33-s + 1.12·41-s − 0.325·43-s + 4/51·51-s + 1.19·57-s − 1.38·59-s − 0.925·67-s + 1.94·73-s − 2/3·75-s − 0.135·81-s + 1.90·83-s − 1.64·89-s + 0.969·97-s + 0.707·99-s + 1.66·107-s + 0.867·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1568} (687, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8273430433\)
\(L(\frac12)\) \(\approx\) \(0.8273430433\)
\(L(2)\) 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 \( 1 + 2 T + p^{2} T^{2} \)
5 \( ( 1 - p T )( 1 + p T ) \)
11 \( 1 + 14 T + p^{2} T^{2} \)
13 \( ( 1 - p T )( 1 + p T ) \)
17 \( 1 + 2 T + p^{2} T^{2} \)
19 \( 1 + 34 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( ( 1 - p T )( 1 + p T ) \)
37 \( ( 1 - p T )( 1 + p T ) \)
41 \( 1 - 46 T + p^{2} T^{2} \)
43 \( 1 + 14 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( 1 + 82 T + p^{2} T^{2} \)
61 \( ( 1 - p T )( 1 + p T ) \)
67 \( 1 + 62 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 142 T + p^{2} T^{2} \)
79 \( ( 1 - p T )( 1 + p T ) \)
83 \( 1 - 158 T + p^{2} T^{2} \)
89 \( 1 + 146 T + p^{2} T^{2} \)
97 \( 1 - 94 T + p^{2} T^{2} \)
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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

−9.152395981110827228439684658366, −8.435746453629335028398027984578, −7.71120773380786649992907513798, −6.64026491016849692559864570405, −6.00070060644862675561151852794, −5.13382945808920307768567833954, −4.43050146077250669005907803669, −3.07989435337194361346381975256, −2.17318398502561784555681740847, −0.50274953242930439082148416241, 0.50274953242930439082148416241, 2.17318398502561784555681740847, 3.07989435337194361346381975256, 4.43050146077250669005907803669, 5.13382945808920307768567833954, 6.00070060644862675561151852794, 6.64026491016849692559864570405, 7.71120773380786649992907513798, 8.435746453629335028398027984578, 9.152395981110827228439684658366

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