Properties

Label 2-1856-1.1-c3-0-124
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  − 5.90·3-s + 14.9·5-s + 6.82·7-s + 7.81·9-s − 18.6·11-s + 39.8·13-s − 88.4·15-s + 19.3·17-s + 2.14·19-s − 40.2·21-s − 103.·23-s + 99.9·25-s + 113.·27-s − 29·29-s − 213.·31-s + 110.·33-s + 102.·35-s + 226.·37-s − 235.·39-s − 191.·41-s + 34.3·43-s + 117.·45-s − 501.·47-s − 296.·49-s − 114.·51-s − 259.·53-s − 280.·55-s + ⋯
L(s)  = 1  − 1.13·3-s + 1.34·5-s + 0.368·7-s + 0.289·9-s − 0.512·11-s + 0.850·13-s − 1.52·15-s + 0.275·17-s + 0.0259·19-s − 0.418·21-s − 0.942·23-s + 0.799·25-s + 0.807·27-s − 0.185·29-s − 1.23·31-s + 0.581·33-s + 0.494·35-s + 1.00·37-s − 0.965·39-s − 0.728·41-s + 0.121·43-s + 0.388·45-s − 1.55·47-s − 0.864·49-s − 0.313·51-s − 0.673·53-s − 0.687·55-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: Trivial
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 \( 1 + 5.90T + 27T^{2} \)
5 \( 1 - 14.9T + 125T^{2} \)
7 \( 1 - 6.82T + 343T^{2} \)
11 \( 1 + 18.6T + 1.33e3T^{2} \)
13 \( 1 - 39.8T + 2.19e3T^{2} \)
17 \( 1 - 19.3T + 4.91e3T^{2} \)
19 \( 1 - 2.14T + 6.85e3T^{2} \)
23 \( 1 + 103.T + 1.21e4T^{2} \)
31 \( 1 + 213.T + 2.97e4T^{2} \)
37 \( 1 - 226.T + 5.06e4T^{2} \)
41 \( 1 + 191.T + 6.89e4T^{2} \)
43 \( 1 - 34.3T + 7.95e4T^{2} \)
47 \( 1 + 501.T + 1.03e5T^{2} \)
53 \( 1 + 259.T + 1.48e5T^{2} \)
59 \( 1 - 280.T + 2.05e5T^{2} \)
61 \( 1 + 81.0T + 2.26e5T^{2} \)
67 \( 1 + 799.T + 3.00e5T^{2} \)
71 \( 1 - 575.T + 3.57e5T^{2} \)
73 \( 1 + 22.5T + 3.89e5T^{2} \)
79 \( 1 - 118.T + 4.93e5T^{2} \)
83 \( 1 - 235.T + 5.71e5T^{2} \)
89 \( 1 - 1.15e3T + 7.04e5T^{2} \)
97 \( 1 - 295.T + 9.12e5T^{2} \)
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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

−8.515224784754819850686776499308, −7.68159371306768354911635666347, −6.49844310181043921275596806976, −6.03230199071975524981079756245, −5.41102050251651284424776949964, −4.73324133219744532869321327661, −3.41321538541475408996948488114, −2.13856921259137118218701907335, −1.29143581509575557557306871291, 0, 1.29143581509575557557306871291, 2.13856921259137118218701907335, 3.41321538541475408996948488114, 4.73324133219744532869321327661, 5.41102050251651284424776949964, 6.03230199071975524981079756245, 6.49844310181043921275596806976, 7.68159371306768354911635666347, 8.515224784754819850686776499308

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