Properties

Label 2-1568-1.1-c1-0-37
Degree $2$
Conductor $1568$
Sign $-1$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s − 5-s − 5.19·11-s − 1.73·15-s − 5·17-s + 1.73·19-s + 1.73·23-s − 4·25-s − 5.19·27-s + 8·29-s − 8.66·31-s − 9·33-s − 5·37-s − 4·41-s − 6.92·43-s + 8.66·47-s − 8.66·51-s − 53-s + 5.19·55-s + 2.99·57-s + 1.73·59-s − 11·61-s + 12.1·67-s + 2.99·69-s + 13.8·71-s − 15·73-s − 6.92·75-s + ⋯
L(s)  = 1  + 1.00·3-s − 0.447·5-s − 1.56·11-s − 0.447·15-s − 1.21·17-s + 0.397·19-s + 0.361·23-s − 0.800·25-s − 1.00·27-s + 1.48·29-s − 1.55·31-s − 1.56·33-s − 0.821·37-s − 0.624·41-s − 1.05·43-s + 1.26·47-s − 1.21·51-s − 0.137·53-s + 0.700·55-s + 0.397·57-s + 0.225·59-s − 1.40·61-s + 1.48·67-s + 0.361·69-s + 1.64·71-s − 1.75·73-s − 0.800·75-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + 8.66T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 6.92T + 43T^{2} \)
47 \( 1 - 8.66T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 - 1.73T + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 - 1.73T + 79T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 + 7T + 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−8.832443402605427668192012600181, −8.293127084651543003213691605111, −7.62473822803488943409818211173, −6.85795088417944299202644685591, −5.62796939912276000558408700129, −4.81553158730289397814414536046, −3.71459672137746628721843940476, −2.87381173586581413683708153789, −2.04633501668482838765722432535, 0, 2.04633501668482838765722432535, 2.87381173586581413683708153789, 3.71459672137746628721843940476, 4.81553158730289397814414536046, 5.62796939912276000558408700129, 6.85795088417944299202644685591, 7.62473822803488943409818211173, 8.293127084651543003213691605111, 8.832443402605427668192012600181

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