Properties

Label 2-1792-1.1-c1-0-10
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  − 1.23·3-s + 1.23·5-s + 7-s − 1.47·9-s + 4·11-s − 1.23·13-s − 1.52·15-s − 2·17-s + 1.23·19-s − 1.23·21-s + 6.47·23-s − 3.47·25-s + 5.52·27-s + 1.52·29-s − 4.94·33-s + 1.23·35-s + 6.47·37-s + 1.52·39-s − 2·41-s + 8.94·43-s − 1.81·45-s − 12.9·47-s + 49-s + 2.47·51-s − 8.94·53-s + 4.94·55-s − 1.52·57-s + ⋯
L(s)  = 1  − 0.713·3-s + 0.552·5-s + 0.377·7-s − 0.490·9-s + 1.20·11-s − 0.342·13-s − 0.394·15-s − 0.485·17-s + 0.283·19-s − 0.269·21-s + 1.34·23-s − 0.694·25-s + 1.06·27-s + 0.283·29-s − 0.860·33-s + 0.208·35-s + 1.06·37-s + 0.244·39-s − 0.312·41-s + 1.36·43-s − 0.271·45-s − 1.88·47-s + 0.142·49-s + 0.346·51-s − 1.22·53-s + 0.666·55-s − 0.202·57-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.543195040\)
\(L(\frac12)\) \(\approx\) \(1.543195040\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.23T + 3T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 - 1.52T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6.47T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 - 9.23T + 59T^{2} \)
61 \( 1 - 1.23T + 61T^{2} \)
67 \( 1 - 1.52T + 67T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 4.94T + 79T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−9.315674211653326294645218594033, −8.632774866063494430817119255835, −7.64581856985470557026708555669, −6.63039384518596305894554324957, −6.16099088283009944142005901262, −5.22451497337483502071372921225, −4.55543857223913567351642932973, −3.34408856227961594942603801579, −2.14882292726780365030061664787, −0.906383485706870872564815430846, 0.906383485706870872564815430846, 2.14882292726780365030061664787, 3.34408856227961594942603801579, 4.55543857223913567351642932973, 5.22451497337483502071372921225, 6.16099088283009944142005901262, 6.63039384518596305894554324957, 7.64581856985470557026708555669, 8.632774866063494430817119255835, 9.315674211653326294645218594033

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