Properties

Label 2-1792-1.1-c1-0-32
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·5-s + 7-s − 3·9-s + 2.82·11-s + 2.82·13-s − 2·17-s + 5.65·19-s − 8·23-s + 3.00·25-s + 5.65·29-s − 8·31-s − 2.82·35-s − 5.65·37-s + 6·41-s + 2.82·43-s + 8.48·45-s − 8·47-s + 49-s − 11.3·53-s − 8.00·55-s − 11.3·59-s − 2.82·61-s − 3·63-s − 8.00·65-s − 8.48·67-s − 8·71-s − 6·73-s + ⋯
L(s)  = 1  − 1.26·5-s + 0.377·7-s − 9-s + 0.852·11-s + 0.784·13-s − 0.485·17-s + 1.29·19-s − 1.66·23-s + 0.600·25-s + 1.05·29-s − 1.43·31-s − 0.478·35-s − 0.929·37-s + 0.937·41-s + 0.431·43-s + 1.26·45-s − 1.16·47-s + 0.142·49-s − 1.55·53-s − 1.07·55-s − 1.47·59-s − 0.362·61-s − 0.377·63-s − 0.992·65-s − 1.03·67-s − 0.949·71-s − 0.702·73-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: \(1\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 2.82T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.799236922178969684769191168121, −8.045245190638781218099304120346, −7.54412029242978308809836179681, −6.45630596817959378454372774453, −5.71630079190482484194003814921, −4.60840781381815747813842834000, −3.80060917499115035392136701315, −3.08307298683603365154117235205, −1.54194435066604884302410940401, 0, 1.54194435066604884302410940401, 3.08307298683603365154117235205, 3.80060917499115035392136701315, 4.60840781381815747813842834000, 5.71630079190482484194003814921, 6.45630596817959378454372774453, 7.54412029242978308809836179681, 8.045245190638781218099304120346, 8.799236922178969684769191168121

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