Properties

Label 2-1856-1.1-c3-0-43
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
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  + 5.04·3-s − 10.9·5-s − 1.11·7-s − 1.58·9-s − 48.5·11-s + 78.4·13-s − 55.2·15-s + 75.2·17-s − 38.4·19-s − 5.62·21-s − 45.6·23-s − 4.98·25-s − 144.·27-s − 29·29-s + 243.·31-s − 244.·33-s + 12.2·35-s + 291.·37-s + 395.·39-s − 178.·41-s − 334.·43-s + 17.3·45-s − 179.·47-s − 341.·49-s + 379.·51-s − 306.·53-s + 532.·55-s + ⋯
L(s)  = 1  + 0.970·3-s − 0.979·5-s − 0.0602·7-s − 0.0586·9-s − 1.33·11-s + 1.67·13-s − 0.950·15-s + 1.07·17-s − 0.464·19-s − 0.0584·21-s − 0.414·23-s − 0.0398·25-s − 1.02·27-s − 0.185·29-s + 1.41·31-s − 1.29·33-s + 0.0590·35-s + 1.29·37-s + 1.62·39-s − 0.681·41-s − 1.18·43-s + 0.0574·45-s − 0.558·47-s − 0.996·49-s + 1.04·51-s − 0.794·53-s + 1.30·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.089188865\)
\(L(\frac12)\) \(\approx\) \(2.089188865\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + 29T \)
good3 \( 1 - 5.04T + 27T^{2} \)
5 \( 1 + 10.9T + 125T^{2} \)
7 \( 1 + 1.11T + 343T^{2} \)
11 \( 1 + 48.5T + 1.33e3T^{2} \)
13 \( 1 - 78.4T + 2.19e3T^{2} \)
17 \( 1 - 75.2T + 4.91e3T^{2} \)
19 \( 1 + 38.4T + 6.85e3T^{2} \)
23 \( 1 + 45.6T + 1.21e4T^{2} \)
31 \( 1 - 243.T + 2.97e4T^{2} \)
37 \( 1 - 291.T + 5.06e4T^{2} \)
41 \( 1 + 178.T + 6.89e4T^{2} \)
43 \( 1 + 334.T + 7.95e4T^{2} \)
47 \( 1 + 179.T + 1.03e5T^{2} \)
53 \( 1 + 306.T + 1.48e5T^{2} \)
59 \( 1 - 725.T + 2.05e5T^{2} \)
61 \( 1 + 166.T + 2.26e5T^{2} \)
67 \( 1 + 5.62T + 3.00e5T^{2} \)
71 \( 1 - 167.T + 3.57e5T^{2} \)
73 \( 1 - 1.19e3T + 3.89e5T^{2} \)
79 \( 1 - 323.T + 4.93e5T^{2} \)
83 \( 1 - 34.1T + 5.71e5T^{2} \)
89 \( 1 - 732.T + 7.04e5T^{2} \)
97 \( 1 - 1.29e3T + 9.12e5T^{2} \)
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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

−8.505867455306871284354318780822, −8.081478450909684737403101787814, −7.82655371929360026802183772983, −6.53812777322301692131563719536, −5.69878703009174893442798340683, −4.63063673986880362060850240366, −3.55698399434187019086605042726, −3.19450001884695662719357575463, −2.03265652399136662840305961938, −0.62904631213729373091509526807, 0.62904631213729373091509526807, 2.03265652399136662840305961938, 3.19450001884695662719357575463, 3.55698399434187019086605042726, 4.63063673986880362060850240366, 5.69878703009174893442798340683, 6.53812777322301692131563719536, 7.82655371929360026802183772983, 8.081478450909684737403101787814, 8.505867455306871284354318780822

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