Properties

Label 2-1840-1.1-c1-0-10
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  − 3.30·3-s + 5-s + 0.302·7-s + 7.90·9-s + 5.30·11-s − 0.302·13-s − 3.30·15-s − 3.90·17-s + 4.90·19-s − 1.00·21-s + 23-s + 25-s − 16.2·27-s + 4.60·29-s − 2.90·31-s − 17.5·33-s + 0.302·35-s + 8·37-s + 1.00·39-s − 9.90·41-s − 5.21·43-s + 7.90·45-s − 4.60·47-s − 6.90·49-s + 12.9·51-s + 3.21·53-s + 5.30·55-s + ⋯
L(s)  = 1  − 1.90·3-s + 0.447·5-s + 0.114·7-s + 2.63·9-s + 1.59·11-s − 0.0839·13-s − 0.852·15-s − 0.947·17-s + 1.12·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 3.11·27-s + 0.855·29-s − 0.522·31-s − 3.04·33-s + 0.0511·35-s + 1.31·37-s + 0.160·39-s − 1.54·41-s − 0.794·43-s + 1.17·45-s − 0.671·47-s − 0.986·49-s + 1.80·51-s + 0.441·53-s + 0.715·55-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

\(L(1)\) \(\approx\) \(1.105343040\)
\(L(\frac12)\) \(\approx\) \(1.105343040\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.30T + 3T^{2} \)
7 \( 1 - 0.302T + 7T^{2} \)
11 \( 1 - 5.30T + 11T^{2} \)
13 \( 1 + 0.302T + 13T^{2} \)
17 \( 1 + 3.90T + 17T^{2} \)
19 \( 1 - 4.90T + 19T^{2} \)
29 \( 1 - 4.60T + 29T^{2} \)
31 \( 1 + 2.90T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 9.90T + 41T^{2} \)
43 \( 1 + 5.21T + 43T^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 - 3.21T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 6.51T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2.69T + 97T^{2} \)
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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

−9.639518072108109095119073302977, −8.559769195233725667887525447222, −7.25137285362637379695327845881, −6.60651191031257657139381908263, −6.18822667182504150839786403064, −5.18966356712349264863638440698, −4.64017857004180320368311422733, −3.62099934720517574066281107058, −1.78870221096584735277916308684, −0.836472660722266300355698285920, 0.836472660722266300355698285920, 1.78870221096584735277916308684, 3.62099934720517574066281107058, 4.64017857004180320368311422733, 5.18966356712349264863638440698, 6.18822667182504150839786403064, 6.60651191031257657139381908263, 7.25137285362637379695327845881, 8.559769195233725667887525447222, 9.639518072108109095119073302977

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