Properties

Label 2-1568-1.1-c1-0-4
Degree $2$
Conductor $1568$
Sign $1$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 3·9-s + 4·13-s − 8·17-s + 11·25-s + 10·29-s + 2·37-s − 8·41-s + 12·45-s + 14·53-s + 12·61-s − 16·65-s + 16·73-s + 9·81-s + 32·85-s + 16·89-s + 8·97-s − 20·101-s − 6·109-s − 14·113-s − 12·117-s + ⋯
L(s)  = 1  − 1.78·5-s − 9-s + 1.10·13-s − 1.94·17-s + 11/5·25-s + 1.85·29-s + 0.328·37-s − 1.24·41-s + 1.78·45-s + 1.92·53-s + 1.53·61-s − 1.98·65-s + 1.87·73-s + 81-s + 3.47·85-s + 1.69·89-s + 0.812·97-s − 1.99·101-s − 0.574·109-s − 1.31·113-s − 1.10·117-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: yes
Arithmetic: yes
Character: $\chi_{1568} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8616555467\)
\(L(\frac12)\) \(\approx\) \(0.8616555467\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.083201307753543769979419428054, −8.396621168864933987238022281800, −8.197509550250461170763039761778, −6.93218317388480289991978591057, −6.43989265358166028466611552196, −5.14312271521888697700030317914, −4.24873327227852142886149189756, −3.55887344410299009293086169964, −2.53443050487252517839586499541, −0.63247991375959544191760657111, 0.63247991375959544191760657111, 2.53443050487252517839586499541, 3.55887344410299009293086169964, 4.24873327227852142886149189756, 5.14312271521888697700030317914, 6.43989265358166028466611552196, 6.93218317388480289991978591057, 8.197509550250461170763039761778, 8.396621168864933987238022281800, 9.083201307753543769979419428054

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