Properties

Label 2-1840-1.1-c1-0-35
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.523·3-s − 5-s − 0.476·7-s − 2.72·9-s + 1.67·11-s + 2.67·13-s − 0.523·15-s + 1.67·17-s − 7.92·19-s − 0.249·21-s − 23-s + 25-s − 3·27-s − 2.20·29-s − 6.77·31-s + 0.878·33-s + 0.476·35-s + 4·37-s + 1.40·39-s + 5.97·41-s + 0.402·43-s + 2.72·45-s − 1.79·47-s − 6.77·49-s + 0.878·51-s − 10.8·53-s − 1.67·55-s + ⋯
L(s)  = 1  + 0.302·3-s − 0.447·5-s − 0.179·7-s − 0.908·9-s + 0.505·11-s + 0.742·13-s − 0.135·15-s + 0.406·17-s − 1.81·19-s − 0.0544·21-s − 0.208·23-s + 0.200·25-s − 0.577·27-s − 0.408·29-s − 1.21·31-s + 0.153·33-s + 0.0804·35-s + 0.657·37-s + 0.224·39-s + 0.933·41-s + 0.0614·43-s + 0.406·45-s − 0.262·47-s − 0.967·49-s + 0.123·51-s − 1.48·53-s − 0.226·55-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: \(1\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.523T + 3T^{2} \)
7 \( 1 + 0.476T + 7T^{2} \)
11 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 - 2.67T + 13T^{2} \)
17 \( 1 - 1.67T + 17T^{2} \)
19 \( 1 + 7.92T + 19T^{2} \)
29 \( 1 + 2.20T + 29T^{2} \)
31 \( 1 + 6.77T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 5.97T + 41T^{2} \)
43 \( 1 - 0.402T + 43T^{2} \)
47 \( 1 + 1.79T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 9.45T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 + 9.97T + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.49T + 83T^{2} \)
89 \( 1 + 3.59T + 89T^{2} \)
97 \( 1 + 6.08T + 97T^{2} \)
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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

−8.831509683789441295958427503674, −8.169610470247004261151053069330, −7.41956921838051250455418890683, −6.29817554689136358218635620268, −5.87338081832233059754996266254, −4.57077346435066343333511912803, −3.76723187475248474017651227626, −2.93381683845095450530871213637, −1.70809905523468441561561219361, 0, 1.70809905523468441561561219361, 2.93381683845095450530871213637, 3.76723187475248474017651227626, 4.57077346435066343333511912803, 5.87338081832233059754996266254, 6.29817554689136358218635620268, 7.41956921838051250455418890683, 8.169610470247004261151053069330, 8.831509683789441295958427503674

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