Properties

Label 2-1856-1.1-c3-0-40
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  + 1.71·3-s − 8.28·5-s − 9.40·7-s − 24.0·9-s + 70.9·11-s + 55.2·13-s − 14.1·15-s − 113.·17-s − 130.·19-s − 16.1·21-s + 119.·23-s − 56.3·25-s − 87.4·27-s − 29·29-s + 152.·31-s + 121.·33-s + 77.9·35-s − 270.·37-s + 94.5·39-s + 368.·41-s + 79.8·43-s + 199.·45-s + 556.·47-s − 254.·49-s − 194.·51-s − 221.·53-s − 588.·55-s + ⋯
L(s)  = 1  + 0.329·3-s − 0.741·5-s − 0.508·7-s − 0.891·9-s + 1.94·11-s + 1.17·13-s − 0.244·15-s − 1.61·17-s − 1.57·19-s − 0.167·21-s + 1.08·23-s − 0.450·25-s − 0.623·27-s − 0.185·29-s + 0.884·31-s + 0.641·33-s + 0.376·35-s − 1.20·37-s + 0.388·39-s + 1.40·41-s + 0.283·43-s + 0.660·45-s + 1.72·47-s − 0.741·49-s − 0.532·51-s − 0.573·53-s − 1.44·55-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

\(L(2)\) \(\approx\) \(1.626550508\)
\(L(\frac12)\) \(\approx\) \(1.626550508\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.71T + 27T^{2} \)
5 \( 1 + 8.28T + 125T^{2} \)
7 \( 1 + 9.40T + 343T^{2} \)
11 \( 1 - 70.9T + 1.33e3T^{2} \)
13 \( 1 - 55.2T + 2.19e3T^{2} \)
17 \( 1 + 113.T + 4.91e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 - 119.T + 1.21e4T^{2} \)
31 \( 1 - 152.T + 2.97e4T^{2} \)
37 \( 1 + 270.T + 5.06e4T^{2} \)
41 \( 1 - 368.T + 6.89e4T^{2} \)
43 \( 1 - 79.8T + 7.95e4T^{2} \)
47 \( 1 - 556.T + 1.03e5T^{2} \)
53 \( 1 + 221.T + 1.48e5T^{2} \)
59 \( 1 + 18.7T + 2.05e5T^{2} \)
61 \( 1 - 298.T + 2.26e5T^{2} \)
67 \( 1 + 516.T + 3.00e5T^{2} \)
71 \( 1 + 568.T + 3.57e5T^{2} \)
73 \( 1 + 660.T + 3.89e5T^{2} \)
79 \( 1 - 35.3T + 4.93e5T^{2} \)
83 \( 1 - 83.5T + 5.71e5T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + 621.T + 9.12e5T^{2} \)
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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

−8.883185872375464818508042967768, −8.414768270306984192677063049244, −7.21666632694291356913529838532, −6.41502111571480716358962705162, −6.02685134516441559264668000800, −4.39623864275331216181877106596, −3.96196240721507376535527428964, −3.07483583254184650917939326145, −1.89160107810222201524562307805, −0.57883646819419286459492220983, 0.57883646819419286459492220983, 1.89160107810222201524562307805, 3.07483583254184650917939326145, 3.96196240721507376535527428964, 4.39623864275331216181877106596, 6.02685134516441559264668000800, 6.41502111571480716358962705162, 7.21666632694291356913529838532, 8.414768270306984192677063049244, 8.883185872375464818508042967768

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