Properties

Label 2-1792-1.1-c1-0-4
Degree $2$
Conductor $1792$
Sign $1$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
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  − 1.08·3-s − 1.08·5-s + 7-s − 1.82·9-s − 5.22·11-s − 6.30·13-s + 1.17·15-s + 3.65·17-s + 4.14·19-s − 1.08·21-s + 1.17·23-s − 3.82·25-s + 5.22·27-s + 8.28·29-s + 5.65·31-s + 5.65·33-s − 1.08·35-s − 2.16·37-s + 6.82·39-s − 7.65·41-s + 5.22·43-s + 1.97·45-s + 8·47-s + 49-s − 3.95·51-s + 4.32·53-s + 5.65·55-s + ⋯
L(s)  = 1  − 0.624·3-s − 0.484·5-s + 0.377·7-s − 0.609·9-s − 1.57·11-s − 1.74·13-s + 0.302·15-s + 0.886·17-s + 0.950·19-s − 0.236·21-s + 0.244·23-s − 0.765·25-s + 1.00·27-s + 1.53·29-s + 1.01·31-s + 0.984·33-s − 0.182·35-s − 0.355·37-s + 1.09·39-s − 1.19·41-s + 0.796·43-s + 0.295·45-s + 1.16·47-s + 0.142·49-s − 0.554·51-s + 0.594·53-s + 0.762·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $1$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8132019882\)
\(L(\frac12)\) \(\approx\) \(0.8132019882\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + 1.08T + 3T^{2} \)
5 \( 1 + 1.08T + 5T^{2} \)
11 \( 1 + 5.22T + 11T^{2} \)
13 \( 1 + 6.30T + 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 - 4.14T + 19T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 2.16T + 37T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 - 5.22T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 4.32T + 53T^{2} \)
59 \( 1 + 6.30T + 59T^{2} \)
61 \( 1 + 7.20T + 61T^{2} \)
67 \( 1 + 7.39T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 0.343T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 5.41T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.384490728801170804529471085586, −8.191108993866912864220992752764, −7.78854599941861435951317265208, −7.04741521069799521059191648291, −5.85826208188774209997500997112, −5.12684869949443878024881240764, −4.70359443630167334754654164063, −3.16463950107020017705630722470, −2.44449392282825867472294226931, −0.60676135026814819524290227587, 0.60676135026814819524290227587, 2.44449392282825867472294226931, 3.16463950107020017705630722470, 4.70359443630167334754654164063, 5.12684869949443878024881240764, 5.85826208188774209997500997112, 7.04741521069799521059191648291, 7.78854599941861435951317265208, 8.191108993866912864220992752764, 9.384490728801170804529471085586

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