Properties

Label 2-1568-1.1-c1-0-19
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.16·3-s − 2.82·5-s + 7.00·9-s + 4.47·11-s − 8.94·15-s − 4.24·17-s + 3.16·19-s + 8.94·23-s + 3.00·25-s + 12.6·27-s − 6·29-s + 6.32·31-s + 14.1·33-s + 2·37-s − 1.41·41-s − 4.47·43-s − 19.7·45-s − 6.32·47-s − 13.4·51-s + 6·53-s − 12.6·55-s + 10.0·57-s + 3.16·59-s + 8.48·61-s + 28.2·69-s + 8.94·71-s + 7.07·73-s + ⋯
L(s)  = 1  + 1.82·3-s − 1.26·5-s + 2.33·9-s + 1.34·11-s − 2.30·15-s − 1.02·17-s + 0.725·19-s + 1.86·23-s + 0.600·25-s + 2.43·27-s − 1.11·29-s + 1.13·31-s + 2.46·33-s + 0.328·37-s − 0.220·41-s − 0.681·43-s − 2.95·45-s − 0.922·47-s − 1.87·51-s + 0.824·53-s − 1.70·55-s + 1.32·57-s + 0.411·59-s + 1.08·61-s + 3.40·69-s + 1.06·71-s + 0.827·73-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.931883303\)
\(L(\frac12)\) \(\approx\) \(2.931883303\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.16T + 3T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 6.32T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 4.47T + 43T^{2} \)
47 \( 1 + 6.32T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 - 7.07T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−9.233661505903103261617081367516, −8.586696222577416729906747946593, −8.039711473485126745942232495043, −7.11957122394869153554876539063, −6.74389614706836314468046805975, −4.89860461005256719269545343341, −3.98354626339970453753901510572, −3.52772005346335463249579889641, −2.56336977921304448822556961718, −1.23629472945942244569028190055, 1.23629472945942244569028190055, 2.56336977921304448822556961718, 3.52772005346335463249579889641, 3.98354626339970453753901510572, 4.89860461005256719269545343341, 6.74389614706836314468046805975, 7.11957122394869153554876539063, 8.039711473485126745942232495043, 8.586696222577416729906747946593, 9.233661505903103261617081367516

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