Properties

Label 2-1792-1.1-c1-0-28
Degree $2$
Conductor $1792$
Sign $1$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s + 7-s + 9-s + 2·11-s − 4·13-s + 8·15-s + 2·17-s − 6·19-s + 2·21-s + 11·25-s − 4·27-s + 8·29-s + 8·31-s + 4·33-s + 4·35-s + 8·37-s − 8·39-s − 10·41-s + 2·43-s + 4·45-s − 8·47-s + 49-s + 4·51-s + 8·55-s − 12·57-s − 10·59-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 2.06·15-s + 0.485·17-s − 1.37·19-s + 0.436·21-s + 11/5·25-s − 0.769·27-s + 1.48·29-s + 1.43·31-s + 0.696·33-s + 0.676·35-s + 1.31·37-s − 1.28·39-s − 1.56·41-s + 0.304·43-s + 0.596·45-s − 1.16·47-s + 1/7·49-s + 0.560·51-s + 1.07·55-s − 1.58·57-s − 1.30·59-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: yes
Arithmetic: yes
Character: $\chi_{1792} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.617445250\)
\(L(\frac12)\) \(\approx\) \(3.617445250\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.318761612207068811842344594127, −8.546625992271387674816885839022, −7.992914711579459891593754479777, −6.74776090668964381863923678873, −6.22556455636945214318257476470, −5.16624989138851774396165460424, −4.36535540120057539344813073391, −2.92567826122973580405986936408, −2.37367773065610120834468856295, −1.45832384904657446684298443518, 1.45832384904657446684298443518, 2.37367773065610120834468856295, 2.92567826122973580405986936408, 4.36535540120057539344813073391, 5.16624989138851774396165460424, 6.22556455636945214318257476470, 6.74776090668964381863923678873, 7.992914711579459891593754479777, 8.546625992271387674816885839022, 9.318761612207068811842344594127

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