Properties

Label 2-1856-1.1-c3-0-119
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  + 2.39·3-s + 3.28·5-s − 33.9·7-s − 21.2·9-s + 14.5·11-s + 86.5·13-s + 7.86·15-s + 102.·17-s − 105.·19-s − 81.3·21-s + 135.·23-s − 114.·25-s − 115.·27-s − 29·29-s − 223.·31-s + 34.9·33-s − 111.·35-s + 239.·37-s + 207.·39-s + 219.·41-s + 18.9·43-s − 69.7·45-s − 147.·47-s + 809.·49-s + 245.·51-s − 613.·53-s + 47.8·55-s + ⋯
L(s)  = 1  + 0.461·3-s + 0.293·5-s − 1.83·7-s − 0.787·9-s + 0.399·11-s + 1.84·13-s + 0.135·15-s + 1.46·17-s − 1.27·19-s − 0.845·21-s + 1.22·23-s − 0.913·25-s − 0.824·27-s − 0.185·29-s − 1.29·31-s + 0.184·33-s − 0.537·35-s + 1.06·37-s + 0.851·39-s + 0.835·41-s + 0.0673·43-s − 0.230·45-s − 0.459·47-s + 2.35·49-s + 0.674·51-s − 1.58·53-s + 0.117·55-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 \( 1 - 2.39T + 27T^{2} \)
5 \( 1 - 3.28T + 125T^{2} \)
7 \( 1 + 33.9T + 343T^{2} \)
11 \( 1 - 14.5T + 1.33e3T^{2} \)
13 \( 1 - 86.5T + 2.19e3T^{2} \)
17 \( 1 - 102.T + 4.91e3T^{2} \)
19 \( 1 + 105.T + 6.85e3T^{2} \)
23 \( 1 - 135.T + 1.21e4T^{2} \)
31 \( 1 + 223.T + 2.97e4T^{2} \)
37 \( 1 - 239.T + 5.06e4T^{2} \)
41 \( 1 - 219.T + 6.89e4T^{2} \)
43 \( 1 - 18.9T + 7.95e4T^{2} \)
47 \( 1 + 147.T + 1.03e5T^{2} \)
53 \( 1 + 613.T + 1.48e5T^{2} \)
59 \( 1 - 184.T + 2.05e5T^{2} \)
61 \( 1 - 13.6T + 2.26e5T^{2} \)
67 \( 1 - 328.T + 3.00e5T^{2} \)
71 \( 1 + 5.15T + 3.57e5T^{2} \)
73 \( 1 + 428.T + 3.89e5T^{2} \)
79 \( 1 - 392.T + 4.93e5T^{2} \)
83 \( 1 + 454.T + 5.71e5T^{2} \)
89 \( 1 + 811.T + 7.04e5T^{2} \)
97 \( 1 + 11.3T + 9.12e5T^{2} \)
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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

−8.652701920207818468837729096055, −7.81544312908712416574292204145, −6.71107835708330829853372525011, −6.06563118126783771475488553755, −5.62929289204703623777309127067, −3.90939656418521474598996414243, −3.44471252786275419054327131019, −2.64409429148702377236338971480, −1.26502932532872807363026550251, 0, 1.26502932532872807363026550251, 2.64409429148702377236338971480, 3.44471252786275419054327131019, 3.90939656418521474598996414243, 5.62929289204703623777309127067, 6.06563118126783771475488553755, 6.71107835708330829853372525011, 7.81544312908712416574292204145, 8.652701920207818468837729096055

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