Properties

Label 2-1848-1.1-c1-0-10
Degree $2$
Conductor $1848$
Sign $1$
Analytic cond. $14.7563$
Root an. cond. $3.84140$
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  + 3-s + 5-s − 7-s + 9-s − 11-s + 3·13-s + 15-s + 7·19-s − 21-s + 6·23-s − 4·25-s + 27-s − 9·29-s − 33-s − 35-s − 3·37-s + 3·39-s + 8·41-s + 10·43-s + 45-s + 3·47-s + 49-s + 6·53-s − 55-s + 7·57-s + 7·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.258·15-s + 1.60·19-s − 0.218·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s − 1.67·29-s − 0.174·33-s − 0.169·35-s − 0.493·37-s + 0.480·39-s + 1.24·41-s + 1.52·43-s + 0.149·45-s + 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s + 0.927·57-s + 0.911·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(14.7563\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1848} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.400035754\)
\(L(\frac12)\) \(\approx\) \(2.400035754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
good5 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.307369660101503848735608048851, −8.604298852143314123591880043985, −7.55348899701599634070161519128, −7.11222711801681130645771244929, −5.87754486690884115680918682368, −5.40433843514449707001844073262, −4.08338088535617776854949082707, −3.29756002757374595064653405790, −2.34847613120160671852182455428, −1.09197528613515654479849884549, 1.09197528613515654479849884549, 2.34847613120160671852182455428, 3.29756002757374595064653405790, 4.08338088535617776854949082707, 5.40433843514449707001844073262, 5.87754486690884115680918682368, 7.11222711801681130645771244929, 7.55348899701599634070161519128, 8.604298852143314123591880043985, 9.307369660101503848735608048851

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