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  + 3.23·3-s + 1.23·5-s + 7.47·9-s + 2.47·11-s − 5.23·13-s + 4.00·15-s + 4.47·17-s − 3.23·19-s + 4·23-s − 3.47·25-s + 14.4·27-s + 4.47·29-s − 6.47·31-s + 8.00·33-s + 4.47·37-s − 16.9·39-s − 0.472·41-s − 2.47·43-s + 9.23·45-s − 1.52·47-s + 14.4·51-s − 10·53-s + 3.05·55-s − 10.4·57-s + 4.76·59-s − 6.76·61-s − 6.47·65-s + ⋯
L(s)  = 1  + 1.86·3-s + 0.552·5-s + 2.49·9-s + 0.745·11-s − 1.45·13-s + 1.03·15-s + 1.08·17-s − 0.742·19-s + 0.834·23-s − 0.694·25-s + 2.78·27-s + 0.830·29-s − 1.16·31-s + 1.39·33-s + 0.735·37-s − 2.71·39-s − 0.0737·41-s − 0.376·43-s + 1.37·45-s − 0.222·47-s + 2.02·51-s − 1.37·53-s + 0.412·55-s − 1.38·57-s + 0.620·59-s − 0.866·61-s − 0.802·65-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

\(L(1)\) \(\approx\) \(3.733043214\)
\(L(\frac12)\) \(\approx\) \(3.733043214\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.23T + 3T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 + 5.23T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + 3.23T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 + 0.472T + 41T^{2} \)
43 \( 1 + 2.47T + 43T^{2} \)
47 \( 1 + 1.52T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 4.76T + 59T^{2} \)
61 \( 1 + 6.76T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 14.9T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 - 4.76T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−9.529362744689139126816148281944, −8.705412889654435934657014059545, −7.889147343207897363008561799069, −7.28831557806710236450435715403, −6.42839971483428242832740500367, −5.11076632810424960799315498835, −4.16898783984654253171752289648, −3.23584617740481828152951930413, −2.41762720435726300677545954570, −1.51418020789488514632355481407, 1.51418020789488514632355481407, 2.41762720435726300677545954570, 3.23584617740481828152951930413, 4.16898783984654253171752289648, 5.11076632810424960799315498835, 6.42839971483428242832740500367, 7.28831557806710236450435715403, 7.889147343207897363008561799069, 8.705412889654435934657014059545, 9.529362744689139126816148281944

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