Properties

Label 2-1856-1.1-c1-0-16
Degree $2$
Conductor $1856$
Sign $-1$
Analytic cond. $14.8202$
Root an. cond. $3.84970$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·5-s − 4·7-s + 6·9-s − 11-s + 3·13-s + 9·15-s + 2·17-s + 4·19-s + 12·21-s + 6·23-s + 4·25-s − 9·27-s + 29-s − 9·31-s + 3·33-s + 12·35-s + 8·37-s − 9·39-s − 8·41-s − 5·43-s − 18·45-s + 7·47-s + 9·49-s − 6·51-s + 5·53-s + 3·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34·5-s − 1.51·7-s + 2·9-s − 0.301·11-s + 0.832·13-s + 2.32·15-s + 0.485·17-s + 0.917·19-s + 2.61·21-s + 1.25·23-s + 4/5·25-s − 1.73·27-s + 0.185·29-s − 1.61·31-s + 0.522·33-s + 2.02·35-s + 1.31·37-s − 1.44·39-s − 1.24·41-s − 0.762·43-s − 2.68·45-s + 1.02·47-s + 9/7·49-s − 0.840·51-s + 0.686·53-s + 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $-1$
Analytic conductor: \(14.8202\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1856,\ (\ :1/2),\ -1)\)

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 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−8.990402515185627546328567296310, −7.74730089520633784498196983232, −7.11856279275544935660492244768, −6.43560003277200267517245298001, −5.67479190582650462878626136824, −4.89577302719386106262640043369, −3.81759314249918795716653221706, −3.20774916762945711068213441442, −0.987975331709208295078797794960, 0, 0.987975331709208295078797794960, 3.20774916762945711068213441442, 3.81759314249918795716653221706, 4.89577302719386106262640043369, 5.67479190582650462878626136824, 6.43560003277200267517245298001, 7.11856279275544935660492244768, 7.74730089520633784498196983232, 8.990402515185627546328567296310

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