Properties

Label 2-1792-1.1-c1-0-26
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  + 2.73·3-s + 2.73·5-s − 7-s + 4.46·9-s − 5.46·11-s + 6.73·13-s + 7.46·15-s + 2·17-s − 1.26·19-s − 2.73·21-s + 3.46·23-s + 2.46·25-s + 3.99·27-s − 1.46·29-s + 4·31-s − 14.9·33-s − 2.73·35-s − 1.46·37-s + 18.3·39-s + 2·41-s − 5.46·43-s + 12.1·45-s + 2.92·47-s + 49-s + 5.46·51-s + 12·53-s − 14.9·55-s + ⋯
L(s)  = 1  + 1.57·3-s + 1.22·5-s − 0.377·7-s + 1.48·9-s − 1.64·11-s + 1.86·13-s + 1.92·15-s + 0.485·17-s − 0.290·19-s − 0.596·21-s + 0.722·23-s + 0.492·25-s + 0.769·27-s − 0.271·29-s + 0.718·31-s − 2.59·33-s − 0.461·35-s − 0.240·37-s + 2.94·39-s + 0.312·41-s − 0.833·43-s + 1.81·45-s + 0.427·47-s + 0.142·49-s + 0.765·51-s + 1.64·53-s − 2.01·55-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

\(L(1)\) \(\approx\) \(3.651144574\)
\(L(\frac12)\) \(\approx\) \(3.651144574\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 3T^{2} \)
5 \( 1 - 2.73T + 5T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 - 6.73T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 1.46T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 1.46T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 5.46T + 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 9.66T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 2.92T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 5.66T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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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

−9.077653491582659354243822839238, −8.615530507070655399749285933414, −7.946863735446870377533255814020, −7.03322669551507082970663008197, −6.01290217464599460849970735609, −5.35585461296207624338836860423, −4.02280686643835579567134157531, −3.07206810596418610267219588385, −2.47151220982796584080128942268, −1.41069666208357436068907091684, 1.41069666208357436068907091684, 2.47151220982796584080128942268, 3.07206810596418610267219588385, 4.02280686643835579567134157531, 5.35585461296207624338836860423, 6.01290217464599460849970735609, 7.03322669551507082970663008197, 7.946863735446870377533255814020, 8.615530507070655399749285933414, 9.077653491582659354243822839238

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