Properties

Degree $2$
Conductor $1568$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.23·3-s − 3.23·5-s − 1.47·9-s − 6.47·11-s − 0.763·13-s + 4.00·15-s − 4.47·17-s + 1.23·19-s + 4·23-s + 5.47·25-s + 5.52·27-s − 4.47·29-s + 2.47·31-s + 8.00·33-s − 4.47·37-s + 0.944·39-s + 8.47·41-s + 6.47·43-s + 4.76·45-s − 10.4·47-s + 5.52·51-s − 10·53-s + 20.9·55-s − 1.52·57-s + 9.23·59-s − 11.2·61-s + 2.47·65-s + ⋯
L(s)  = 1  − 0.713·3-s − 1.44·5-s − 0.490·9-s − 1.95·11-s − 0.211·13-s + 1.03·15-s − 1.08·17-s + 0.283·19-s + 0.834·23-s + 1.09·25-s + 1.06·27-s − 0.830·29-s + 0.444·31-s + 1.39·33-s − 0.735·37-s + 0.151·39-s + 1.32·41-s + 0.986·43-s + 0.710·45-s − 1.52·47-s + 0.774·51-s − 1.37·53-s + 2.82·55-s − 0.202·57-s + 1.20·59-s − 1.43·61-s + 0.306·65-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: $1$
Motivic weight: \(1\)
Character: $\chi_{1568} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3546095847\)
\(L(\frac12)\) \(\approx\) \(0.3546095847\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.23T + 3T^{2} \)
5 \( 1 + 3.23T + 5T^{2} \)
11 \( 1 + 6.47T + 11T^{2} \)
13 \( 1 + 0.763T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 - 8.47T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 9.23T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 - 2.94T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−9.381330686401094257156954132096, −8.423572559963947702829390767235, −7.81510927903905389766787600569, −7.17800403749017692338791301821, −6.13871497065221366604902274811, −5.12614524957281905504214628061, −4.64974042908500062617369353778, −3.40510413960881198425224142312, −2.50828880986518194453449779751, −0.40506550031819905158841571449, 0.40506550031819905158841571449, 2.50828880986518194453449779751, 3.40510413960881198425224142312, 4.64974042908500062617369353778, 5.12614524957281905504214628061, 6.13871497065221366604902274811, 7.17800403749017692338791301821, 7.81510927903905389766787600569, 8.423572559963947702829390767235, 9.381330686401094257156954132096

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