Properties

Label 2-1856-1.1-c3-0-63
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  + 4.03·3-s − 15.2·5-s + 33.3·7-s − 10.7·9-s + 4.44·11-s + 38.9·13-s − 61.5·15-s − 18.1·17-s − 75.1·19-s + 134.·21-s + 187.·23-s + 107.·25-s − 152.·27-s − 29·29-s + 68.3·31-s + 17.9·33-s − 508.·35-s − 44.9·37-s + 156.·39-s + 299.·41-s + 138.·43-s + 163.·45-s − 531.·47-s + 769.·49-s − 73.0·51-s + 242.·53-s − 67.8·55-s + ⋯
L(s)  = 1  + 0.775·3-s − 1.36·5-s + 1.80·7-s − 0.397·9-s + 0.121·11-s + 0.830·13-s − 1.05·15-s − 0.258·17-s − 0.907·19-s + 1.39·21-s + 1.70·23-s + 0.862·25-s − 1.08·27-s − 0.185·29-s + 0.396·31-s + 0.0945·33-s − 2.45·35-s − 0.199·37-s + 0.644·39-s + 1.13·41-s + 0.490·43-s + 0.542·45-s − 1.64·47-s + 2.24·49-s − 0.200·51-s + 0.629·53-s − 0.166·55-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

\(L(2)\) \(\approx\) \(2.740743617\)
\(L(\frac12)\) \(\approx\) \(2.740743617\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.03T + 27T^{2} \)
5 \( 1 + 15.2T + 125T^{2} \)
7 \( 1 - 33.3T + 343T^{2} \)
11 \( 1 - 4.44T + 1.33e3T^{2} \)
13 \( 1 - 38.9T + 2.19e3T^{2} \)
17 \( 1 + 18.1T + 4.91e3T^{2} \)
19 \( 1 + 75.1T + 6.85e3T^{2} \)
23 \( 1 - 187.T + 1.21e4T^{2} \)
31 \( 1 - 68.3T + 2.97e4T^{2} \)
37 \( 1 + 44.9T + 5.06e4T^{2} \)
41 \( 1 - 299.T + 6.89e4T^{2} \)
43 \( 1 - 138.T + 7.95e4T^{2} \)
47 \( 1 + 531.T + 1.03e5T^{2} \)
53 \( 1 - 242.T + 1.48e5T^{2} \)
59 \( 1 + 500.T + 2.05e5T^{2} \)
61 \( 1 - 325.T + 2.26e5T^{2} \)
67 \( 1 - 263.T + 3.00e5T^{2} \)
71 \( 1 - 726.T + 3.57e5T^{2} \)
73 \( 1 - 851.T + 3.89e5T^{2} \)
79 \( 1 - 85.0T + 4.93e5T^{2} \)
83 \( 1 + 914.T + 5.71e5T^{2} \)
89 \( 1 + 1.07e3T + 7.04e5T^{2} \)
97 \( 1 + 662.T + 9.12e5T^{2} \)
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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

−8.529109842572756246340881460591, −8.246532293686133285522798479794, −7.63713675835388681667104273872, −6.74728111981963345009592942989, −5.47717309310877553175151420639, −4.57080040298765080274333379983, −3.95198572847476000915725640952, −3.00103025955831425961186159401, −1.90068112808130427246010438757, −0.76241098611340975065378953799, 0.76241098611340975065378953799, 1.90068112808130427246010438757, 3.00103025955831425961186159401, 3.95198572847476000915725640952, 4.57080040298765080274333379983, 5.47717309310877553175151420639, 6.74728111981963345009592942989, 7.63713675835388681667104273872, 8.246532293686133285522798479794, 8.529109842572756246340881460591

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