Properties

Label 2-1840-1.1-c1-0-11
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.302·3-s + 5-s − 3.30·7-s − 2.90·9-s + 1.69·11-s + 3.30·13-s + 0.302·15-s + 6.90·17-s − 5.90·19-s − 1.00·21-s + 23-s + 25-s − 1.78·27-s − 2.60·29-s + 7.90·31-s + 0.513·33-s − 3.30·35-s + 8·37-s + 1.00·39-s + 0.908·41-s + 9.21·43-s − 2.90·45-s + 2.60·47-s + 3.90·49-s + 2.09·51-s − 11.2·53-s + 1.69·55-s + ⋯
L(s)  = 1  + 0.174·3-s + 0.447·5-s − 1.24·7-s − 0.969·9-s + 0.511·11-s + 0.916·13-s + 0.0781·15-s + 1.67·17-s − 1.35·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.344·27-s − 0.483·29-s + 1.42·31-s + 0.0894·33-s − 0.558·35-s + 1.31·37-s + 0.160·39-s + 0.141·41-s + 1.40·43-s − 0.433·45-s + 0.380·47-s + 0.558·49-s + 0.292·51-s − 1.53·53-s + 0.228·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.712297062\)
\(L(\frac12)\) \(\approx\) \(1.712297062\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 - 0.302T + 3T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 - 1.69T + 11T^{2} \)
13 \( 1 - 3.30T + 13T^{2} \)
17 \( 1 - 6.90T + 17T^{2} \)
19 \( 1 + 5.90T + 19T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 - 7.90T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 0.908T + 41T^{2} \)
43 \( 1 - 9.21T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 3.39T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + 5.81T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 6.30T + 97T^{2} \)
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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

−9.356191689125399719788492406650, −8.511630220375591229806182939849, −7.83763763444015083861030807599, −6.54854135794469658397589842182, −6.18235394571940089081014721812, −5.45130256126043818109353580816, −4.07045182124824515884879004844, −3.28323056710320615391656462273, −2.44607867909984537662334278063, −0.888337481180660831928006609144, 0.888337481180660831928006609144, 2.44607867909984537662334278063, 3.28323056710320615391656462273, 4.07045182124824515884879004844, 5.45130256126043818109353580816, 6.18235394571940089081014721812, 6.54854135794469658397589842182, 7.83763763444015083861030807599, 8.511630220375591229806182939849, 9.356191689125399719788492406650

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