Properties

Label 2-1840-1.1-c3-0-30
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $108.563$
Root an. cond. $10.4193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.62·3-s + 5·5-s − 20.4·7-s − 13.8·9-s − 36.2·11-s − 10.6·13-s + 18.1·15-s − 72.1·17-s + 140.·19-s − 74.1·21-s + 23·23-s + 25·25-s − 148.·27-s − 1.78·29-s + 116.·31-s − 131.·33-s − 102.·35-s + 54.9·37-s − 38.5·39-s + 156.·41-s − 367.·43-s − 69.1·45-s + 20.8·47-s + 74.9·49-s − 261.·51-s − 208.·53-s − 181.·55-s + ⋯
L(s)  = 1  + 0.698·3-s + 0.447·5-s − 1.10·7-s − 0.512·9-s − 0.994·11-s − 0.226·13-s + 0.312·15-s − 1.02·17-s + 1.69·19-s − 0.770·21-s + 0.208·23-s + 0.200·25-s − 1.05·27-s − 0.0114·29-s + 0.673·31-s − 0.694·33-s − 0.493·35-s + 0.244·37-s − 0.158·39-s + 0.595·41-s − 1.30·43-s − 0.229·45-s + 0.0647·47-s + 0.218·49-s − 0.718·51-s − 0.541·53-s − 0.444·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(108.563\)
Root analytic conductor: \(10.4193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.888341997\)
\(L(\frac12)\) \(\approx\) \(1.888341997\)
\(L(\frac{5}{2})\) 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 \( 1 - 5T \)
23 \( 1 - 23T \)
good3 \( 1 - 3.62T + 27T^{2} \)
7 \( 1 + 20.4T + 343T^{2} \)
11 \( 1 + 36.2T + 1.33e3T^{2} \)
13 \( 1 + 10.6T + 2.19e3T^{2} \)
17 \( 1 + 72.1T + 4.91e3T^{2} \)
19 \( 1 - 140.T + 6.85e3T^{2} \)
29 \( 1 + 1.78T + 2.43e4T^{2} \)
31 \( 1 - 116.T + 2.97e4T^{2} \)
37 \( 1 - 54.9T + 5.06e4T^{2} \)
41 \( 1 - 156.T + 6.89e4T^{2} \)
43 \( 1 + 367.T + 7.95e4T^{2} \)
47 \( 1 - 20.8T + 1.03e5T^{2} \)
53 \( 1 + 208.T + 1.48e5T^{2} \)
59 \( 1 - 856.T + 2.05e5T^{2} \)
61 \( 1 - 733.T + 2.26e5T^{2} \)
67 \( 1 - 738.T + 3.00e5T^{2} \)
71 \( 1 - 193.T + 3.57e5T^{2} \)
73 \( 1 + 1.02e3T + 3.89e5T^{2} \)
79 \( 1 - 64.2T + 4.93e5T^{2} \)
83 \( 1 - 924.T + 5.71e5T^{2} \)
89 \( 1 - 83.0T + 7.04e5T^{2} \)
97 \( 1 - 507.T + 9.12e5T^{2} \)
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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

−8.913904493921170645865951423429, −8.213482746408935898119224542925, −7.34161333964794655376498979118, −6.55717673049049946942282471540, −5.66130824775168598129183436331, −4.92805204857627157877342434032, −3.57818580253063440030257793354, −2.88539114932523666048344906882, −2.20604864472483336860147177697, −0.58795270005524117642425162926, 0.58795270005524117642425162926, 2.20604864472483336860147177697, 2.88539114932523666048344906882, 3.57818580253063440030257793354, 4.92805204857627157877342434032, 5.66130824775168598129183436331, 6.55717673049049946942282471540, 7.34161333964794655376498979118, 8.213482746408935898119224542925, 8.913904493921170645865951423429

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