Properties

Label 2-1792-1.1-c1-0-11
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  + 0.732·3-s − 0.732·5-s + 7-s − 2.46·9-s − 1.46·11-s + 3.26·13-s − 0.535·15-s + 2·17-s + 4.73·19-s + 0.732·21-s + 3.46·23-s − 4.46·25-s − 4·27-s + 5.46·29-s − 4·31-s − 1.07·33-s − 0.732·35-s + 5.46·37-s + 2.39·39-s + 2·41-s − 1.46·43-s + 1.80·45-s + 10.9·47-s + 49-s + 1.46·51-s + 12·53-s + 1.07·55-s + ⋯
L(s)  = 1  + 0.422·3-s − 0.327·5-s + 0.377·7-s − 0.821·9-s − 0.441·11-s + 0.906·13-s − 0.138·15-s + 0.485·17-s + 1.08·19-s + 0.159·21-s + 0.722·23-s − 0.892·25-s − 0.769·27-s + 1.01·29-s − 0.718·31-s − 0.186·33-s − 0.123·35-s + 0.898·37-s + 0.383·39-s + 0.312·41-s − 0.223·43-s + 0.268·45-s + 1.59·47-s + 0.142·49-s + 0.205·51-s + 1.64·53-s + 0.144·55-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

\(L(1)\) \(\approx\) \(1.919550154\)
\(L(\frac12)\) \(\approx\) \(1.919550154\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732T + 3T^{2} \)
5 \( 1 + 0.732T + 5T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
13 \( 1 - 3.26T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 5.46T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 5.46T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 7.66T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 + 2.92T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 4.92T + 97T^{2} \)
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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

−9.082212968115080639358554541556, −8.495185751396153071299639606507, −7.79867310516911357133777532705, −7.12412055238422234496933349023, −5.85841686944782568620606814940, −5.38064094286548482556086175334, −4.16159996138097319539441839007, −3.30822418138074584916558933490, −2.42540600721553076586501107411, −0.955944972784133889138847899236, 0.955944972784133889138847899236, 2.42540600721553076586501107411, 3.30822418138074584916558933490, 4.16159996138097319539441839007, 5.38064094286548482556086175334, 5.85841686944782568620606814940, 7.12412055238422234496933349023, 7.79867310516911357133777532705, 8.495185751396153071299639606507, 9.082212968115080639358554541556

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