Properties

Label 2-1792-1.1-c1-0-41
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  + 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: $\chi_{1792} (1, \cdot )$
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 \( 1 - 1.23T + 3T^{2} \)
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

−8.639317914738830253527066503883, −8.226918823380681324938839163272, −7.53112219730028550957823463418, −6.63506693508790279560739365532, −5.48289574234310409620671623617, −4.79589178773493400445909396160, −3.63971843644313591057156594938, −2.91077978729400868052941694003, −1.86105559406579022162662483233, 0, 1.86105559406579022162662483233, 2.91077978729400868052941694003, 3.63971843644313591057156594938, 4.79589178773493400445909396160, 5.48289574234310409620671623617, 6.63506693508790279560739365532, 7.53112219730028550957823463418, 8.226918823380681324938839163272, 8.639317914738830253527066503883

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