Properties

Label 2-1840-1.1-c1-0-0
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  − 1.93·3-s − 5-s − 2.38·7-s + 0.735·9-s − 5.33·11-s − 4.53·13-s + 1.93·15-s + 1.81·17-s − 7.00·19-s + 4.60·21-s + 23-s + 25-s + 4.37·27-s − 0.118·29-s + 0.884·31-s + 10.3·33-s + 2.38·35-s + 7.51·37-s + 8.77·39-s − 1.45·41-s − 0.735·45-s − 10.4·47-s − 1.32·49-s − 3.50·51-s − 9.42·53-s + 5.33·55-s + 13.5·57-s + ⋯
L(s)  = 1  − 1.11·3-s − 0.447·5-s − 0.900·7-s + 0.245·9-s − 1.60·11-s − 1.25·13-s + 0.499·15-s + 0.440·17-s − 1.60·19-s + 1.00·21-s + 0.208·23-s + 0.200·25-s + 0.842·27-s − 0.0219·29-s + 0.158·31-s + 1.79·33-s + 0.402·35-s + 1.23·37-s + 1.40·39-s − 0.226·41-s − 0.109·45-s − 1.52·47-s − 0.189·49-s − 0.491·51-s − 1.29·53-s + 0.719·55-s + 1.79·57-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

\(L(1)\) \(\approx\) \(0.2478813656\)
\(L(\frac12)\) \(\approx\) \(0.2478813656\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93T + 3T^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
11 \( 1 + 5.33T + 11T^{2} \)
13 \( 1 + 4.53T + 13T^{2} \)
17 \( 1 - 1.81T + 17T^{2} \)
19 \( 1 + 7.00T + 19T^{2} \)
29 \( 1 + 0.118T + 29T^{2} \)
31 \( 1 - 0.884T + 31T^{2} \)
37 \( 1 - 7.51T + 37T^{2} \)
41 \( 1 + 1.45T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 9.42T + 53T^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 + 2.80T + 61T^{2} \)
67 \( 1 - 3.11T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 6.80T + 79T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 2.89T + 89T^{2} \)
97 \( 1 + 1.97T + 97T^{2} \)
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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

−9.491403231926375014812525892123, −8.257065013165311266685521909733, −7.69066969321607573581739248795, −6.66916328489374909934527829284, −6.12595048408386719338227348832, −5.10963416088376515193759381939, −4.64863388003871234103167066276, −3.25875493881566695097223488837, −2.37603137320533751218488741797, −0.32957100908932285842405515070, 0.32957100908932285842405515070, 2.37603137320533751218488741797, 3.25875493881566695097223488837, 4.64863388003871234103167066276, 5.10963416088376515193759381939, 6.12595048408386719338227348832, 6.66916328489374909934527829284, 7.69066969321607573581739248795, 8.257065013165311266685521909733, 9.491403231926375014812525892123

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