Properties

Label 2-1792-1.1-c1-0-43
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  + 2.61·3-s − 2.61·5-s − 7-s + 3.82·9-s − 2.16·11-s − 0.448·13-s − 6.82·15-s − 7.65·17-s + 4.77·19-s − 2.61·21-s − 6.82·23-s + 1.82·25-s + 2.16·27-s − 9.55·29-s + 5.65·31-s − 5.65·33-s + 2.61·35-s − 5.22·37-s − 1.17·39-s + 3.65·41-s + 2.16·43-s − 10.0·45-s − 8·47-s + 49-s − 20.0·51-s + 10.4·53-s + 5.65·55-s + ⋯
L(s)  = 1  + 1.50·3-s − 1.16·5-s − 0.377·7-s + 1.27·9-s − 0.652·11-s − 0.124·13-s − 1.76·15-s − 1.85·17-s + 1.09·19-s − 0.570·21-s − 1.42·23-s + 0.365·25-s + 0.416·27-s − 1.77·29-s + 1.01·31-s − 0.984·33-s + 0.441·35-s − 0.859·37-s − 0.187·39-s + 0.571·41-s + 0.330·43-s − 1.49·45-s − 1.16·47-s + 0.142·49-s − 2.80·51-s + 1.43·53-s + 0.762·55-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: \(1\)
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 \( 1 - 2.61T + 3T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
11 \( 1 + 2.16T + 11T^{2} \)
13 \( 1 + 0.448T + 13T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 - 4.77T + 19T^{2} \)
23 \( 1 + 6.82T + 23T^{2} \)
29 \( 1 + 9.55T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 5.22T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 - 2.16T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 0.448T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 3.06T + 67T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.710351195868458156783896321554, −8.163648549799035032896290134777, −7.50299538789660326894609640129, −6.89776450129221367628295002733, −5.59326452443745126862025413529, −4.30748513064606110658485007892, −3.80052614364130295761363185210, −2.88224036748915104976257351764, −2.03196844387901075673338940956, 0, 2.03196844387901075673338940956, 2.88224036748915104976257351764, 3.80052614364130295761363185210, 4.30748513064606110658485007892, 5.59326452443745126862025413529, 6.89776450129221367628295002733, 7.50299538789660326894609640129, 8.163648549799035032896290134777, 8.710351195868458156783896321554

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