Properties

Label 2-1840-1.1-c1-0-2
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  − 1.56·3-s + 5-s − 4.06·7-s − 0.561·9-s − 2.65·11-s − 5.91·13-s − 1.56·15-s − 3.40·17-s + 2.65·19-s + 6.34·21-s + 23-s + 25-s + 5.56·27-s + 5.84·29-s + 9.31·31-s + 4.15·33-s − 4.06·35-s − 4.18·37-s + 9.22·39-s + 2.15·41-s − 2.34·43-s − 0.561·45-s − 0.242·47-s + 9.52·49-s + 5.31·51-s + 9.03·53-s − 2.65·55-s + ⋯
L(s)  = 1  − 0.901·3-s + 0.447·5-s − 1.53·7-s − 0.187·9-s − 0.801·11-s − 1.63·13-s − 0.403·15-s − 0.826·17-s + 0.610·19-s + 1.38·21-s + 0.208·23-s + 0.200·25-s + 1.07·27-s + 1.08·29-s + 1.67·31-s + 0.722·33-s − 0.687·35-s − 0.687·37-s + 1.47·39-s + 0.336·41-s − 0.358·43-s − 0.0837·45-s − 0.0353·47-s + 1.36·49-s + 0.744·51-s + 1.24·53-s − 0.358·55-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

\(L(1)\) \(\approx\) \(0.5825477789\)
\(L(\frac12)\) \(\approx\) \(0.5825477789\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 3T^{2} \)
7 \( 1 + 4.06T + 7T^{2} \)
11 \( 1 + 2.65T + 11T^{2} \)
13 \( 1 + 5.91T + 13T^{2} \)
17 \( 1 + 3.40T + 17T^{2} \)
19 \( 1 - 2.65T + 19T^{2} \)
29 \( 1 - 5.84T + 29T^{2} \)
31 \( 1 - 9.31T + 31T^{2} \)
37 \( 1 + 4.18T + 37T^{2} \)
41 \( 1 - 2.15T + 41T^{2} \)
43 \( 1 + 2.34T + 43T^{2} \)
47 \( 1 + 0.242T + 47T^{2} \)
53 \( 1 - 9.03T + 53T^{2} \)
59 \( 1 + 14.4T + 59T^{2} \)
61 \( 1 - 2.46T + 61T^{2} \)
67 \( 1 - 7.75T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 2.08T + 73T^{2} \)
79 \( 1 - 4.15T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 9.35T + 89T^{2} \)
97 \( 1 - 3.47T + 97T^{2} \)
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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

−9.448355821962019188306594959699, −8.588764979443132934831048502225, −7.44132510671285786635463458469, −6.66928554553747075411912073914, −6.13840300059397530312556960280, −5.22888143474010570175501226955, −4.61696772601981525544348457081, −3.05600560027485522233253335473, −2.50598780698225562346795386547, −0.50520391270906633699124270057, 0.50520391270906633699124270057, 2.50598780698225562346795386547, 3.05600560027485522233253335473, 4.61696772601981525544348457081, 5.22888143474010570175501226955, 6.13840300059397530312556960280, 6.66928554553747075411912073914, 7.44132510671285786635463458469, 8.588764979443132934831048502225, 9.448355821962019188306594959699

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