Properties

Label 2-1792-1.1-c1-0-0
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  − 0.936·3-s − 3.33·5-s − 7-s − 2.12·9-s − 4.27·11-s − 3.33·13-s + 3.12·15-s + 2·17-s − 0.936·19-s + 0.936·21-s − 3.12·23-s + 6.12·25-s + 4.79·27-s − 1.87·29-s − 6.24·31-s + 3.99·33-s + 3.33·35-s − 1.87·37-s + 3.12·39-s + 12.2·41-s − 4.27·43-s + 7.08·45-s + 49-s − 1.87·51-s + 8.54·53-s + 14.2·55-s + 0.876·57-s + ⋯
L(s)  = 1  − 0.540·3-s − 1.49·5-s − 0.377·7-s − 0.707·9-s − 1.28·11-s − 0.924·13-s + 0.806·15-s + 0.485·17-s − 0.214·19-s + 0.204·21-s − 0.651·23-s + 1.22·25-s + 0.923·27-s − 0.347·29-s − 1.12·31-s + 0.696·33-s + 0.563·35-s − 0.307·37-s + 0.500·39-s + 1.91·41-s − 0.651·43-s + 1.05·45-s + 0.142·49-s − 0.262·51-s + 1.17·53-s + 1.92·55-s + 0.116·57-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

\(L(1)\) \(\approx\) \(0.3078850050\)
\(L(\frac12)\) \(\approx\) \(0.3078850050\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.936T + 3T^{2} \)
5 \( 1 + 3.33T + 5T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + 3.33T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 0.936T + 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 + 1.87T + 29T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 + 1.87T + 37T^{2} \)
41 \( 1 - 12.2T + 41T^{2} \)
43 \( 1 + 4.27T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 8.54T + 53T^{2} \)
59 \( 1 - 7.60T + 59T^{2} \)
61 \( 1 + 3.33T + 61T^{2} \)
67 \( 1 + 15.7T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 9.47T + 83T^{2} \)
89 \( 1 + 0.246T + 89T^{2} \)
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

−9.218856485195048762553779362359, −8.306063520283419449521538100694, −7.65921843034752959170029649385, −7.14012749393927763155695626067, −5.91991487950114033278448733696, −5.25172061616993512094772609216, −4.33911346829486591503435058394, −3.37586428681008076109563540564, −2.47521103627974456287929866930, −0.36139886898797756619735050082, 0.36139886898797756619735050082, 2.47521103627974456287929866930, 3.37586428681008076109563540564, 4.33911346829486591503435058394, 5.25172061616993512094772609216, 5.91991487950114033278448733696, 7.14012749393927763155695626067, 7.65921843034752959170029649385, 8.306063520283419449521538100694, 9.218856485195048762553779362359

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