Properties

Label 2-1856-1.1-c3-0-8
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  − 0.323·3-s + 5.78·5-s − 34.7·7-s − 26.8·9-s − 18.6·11-s − 38.6·13-s − 1.87·15-s − 101.·17-s − 1.73·19-s + 11.2·21-s − 24.4·23-s − 91.5·25-s + 17.4·27-s − 29·29-s − 186.·31-s + 6.04·33-s − 201.·35-s + 339.·37-s + 12.5·39-s + 34.8·41-s − 483.·43-s − 155.·45-s − 266.·47-s + 866.·49-s + 32.7·51-s − 148.·53-s − 107.·55-s + ⋯
L(s)  = 1  − 0.0623·3-s + 0.517·5-s − 1.87·7-s − 0.996·9-s − 0.511·11-s − 0.825·13-s − 0.0322·15-s − 1.44·17-s − 0.0209·19-s + 0.117·21-s − 0.221·23-s − 0.732·25-s + 0.124·27-s − 0.185·29-s − 1.07·31-s + 0.0318·33-s − 0.971·35-s + 1.50·37-s + 0.0514·39-s + 0.132·41-s − 1.71·43-s − 0.515·45-s − 0.827·47-s + 2.52·49-s + 0.0900·51-s − 0.385·53-s − 0.264·55-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

\(L(2)\) \(\approx\) \(0.2139399242\)
\(L(\frac12)\) \(\approx\) \(0.2139399242\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.323T + 27T^{2} \)
5 \( 1 - 5.78T + 125T^{2} \)
7 \( 1 + 34.7T + 343T^{2} \)
11 \( 1 + 18.6T + 1.33e3T^{2} \)
13 \( 1 + 38.6T + 2.19e3T^{2} \)
17 \( 1 + 101.T + 4.91e3T^{2} \)
19 \( 1 + 1.73T + 6.85e3T^{2} \)
23 \( 1 + 24.4T + 1.21e4T^{2} \)
31 \( 1 + 186.T + 2.97e4T^{2} \)
37 \( 1 - 339.T + 5.06e4T^{2} \)
41 \( 1 - 34.8T + 6.89e4T^{2} \)
43 \( 1 + 483.T + 7.95e4T^{2} \)
47 \( 1 + 266.T + 1.03e5T^{2} \)
53 \( 1 + 148.T + 1.48e5T^{2} \)
59 \( 1 + 6.98T + 2.05e5T^{2} \)
61 \( 1 + 38.0T + 2.26e5T^{2} \)
67 \( 1 + 544.T + 3.00e5T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 - 688.T + 3.89e5T^{2} \)
79 \( 1 + 424.T + 4.93e5T^{2} \)
83 \( 1 + 1.24e3T + 5.71e5T^{2} \)
89 \( 1 + 174.T + 7.04e5T^{2} \)
97 \( 1 - 33.7T + 9.12e5T^{2} \)
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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

−9.119548809500377691747511150816, −8.162273851373721648205121298399, −7.13551041525956969536752153123, −6.37850527462455735106109529982, −5.86737550776657947742797263881, −4.93195360826196895824372178548, −3.72607771175642482123666023327, −2.83081031319681662431535431038, −2.15715212984364516338003840908, −0.19689741495151284776369080391, 0.19689741495151284776369080391, 2.15715212984364516338003840908, 2.83081031319681662431535431038, 3.72607771175642482123666023327, 4.93195360826196895824372178548, 5.86737550776657947742797263881, 6.37850527462455735106109529982, 7.13551041525956969536752153123, 8.162273851373721648205121298399, 9.119548809500377691747511150816

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