Properties

Label 2-1856-1.1-c3-0-154
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.46·3-s − 2.14·5-s + 20.3·7-s + 14.7·9-s − 52.0·11-s − 7.04·13-s − 13.8·15-s + 28.7·17-s − 76.4·19-s + 131.·21-s + 59.7·23-s − 120.·25-s − 79.0·27-s + 29·29-s − 3.25·31-s − 336.·33-s − 43.6·35-s − 150.·37-s − 45.5·39-s − 92.3·41-s + 100.·43-s − 31.6·45-s + 324.·47-s + 71.4·49-s + 185.·51-s − 374.·53-s + 111.·55-s + ⋯
L(s)  = 1  + 1.24·3-s − 0.191·5-s + 1.09·7-s + 0.547·9-s − 1.42·11-s − 0.150·13-s − 0.238·15-s + 0.410·17-s − 0.922·19-s + 1.36·21-s + 0.541·23-s − 0.963·25-s − 0.563·27-s + 0.185·29-s − 0.0188·31-s − 1.77·33-s − 0.210·35-s − 0.669·37-s − 0.186·39-s − 0.351·41-s + 0.357·43-s − 0.104·45-s + 1.00·47-s + 0.208·49-s + 0.510·51-s − 0.971·53-s + 0.273·55-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: \(1\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 6.46T + 27T^{2} \)
5 \( 1 + 2.14T + 125T^{2} \)
7 \( 1 - 20.3T + 343T^{2} \)
11 \( 1 + 52.0T + 1.33e3T^{2} \)
13 \( 1 + 7.04T + 2.19e3T^{2} \)
17 \( 1 - 28.7T + 4.91e3T^{2} \)
19 \( 1 + 76.4T + 6.85e3T^{2} \)
23 \( 1 - 59.7T + 1.21e4T^{2} \)
31 \( 1 + 3.25T + 2.97e4T^{2} \)
37 \( 1 + 150.T + 5.06e4T^{2} \)
41 \( 1 + 92.3T + 6.89e4T^{2} \)
43 \( 1 - 100.T + 7.95e4T^{2} \)
47 \( 1 - 324.T + 1.03e5T^{2} \)
53 \( 1 + 374.T + 1.48e5T^{2} \)
59 \( 1 + 489.T + 2.05e5T^{2} \)
61 \( 1 + 221.T + 2.26e5T^{2} \)
67 \( 1 - 427.T + 3.00e5T^{2} \)
71 \( 1 + 898.T + 3.57e5T^{2} \)
73 \( 1 + 1.08e3T + 3.89e5T^{2} \)
79 \( 1 + 798.T + 4.93e5T^{2} \)
83 \( 1 - 436.T + 5.71e5T^{2} \)
89 \( 1 - 456.T + 7.04e5T^{2} \)
97 \( 1 - 803.T + 9.12e5T^{2} \)
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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

−8.462529586892747503017197690174, −7.73019032236109896470564432499, −7.44499635353425335410055206997, −6.00725321931339914658241704232, −5.08622840736103304014441220342, −4.31446204667673888821781915795, −3.24658036428366790454104490919, −2.44448436074356498566047688829, −1.62754872869830660272071734016, 0, 1.62754872869830660272071734016, 2.44448436074356498566047688829, 3.24658036428366790454104490919, 4.31446204667673888821781915795, 5.08622840736103304014441220342, 6.00725321931339914658241704232, 7.44499635353425335410055206997, 7.73019032236109896470564432499, 8.462529586892747503017197690174

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