Properties

Label 2-1856-1.1-c3-0-128
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9.87·3-s + 16.8·5-s + 5.21·7-s + 70.6·9-s + 8.55·11-s + 11.3·13-s − 166.·15-s + 68.4·17-s − 6.93·19-s − 51.5·21-s − 132.·23-s + 157.·25-s − 430.·27-s + 29·29-s − 0.419·31-s − 84.4·33-s + 87.8·35-s − 395.·37-s − 112.·39-s − 447.·41-s − 184.·43-s + 1.18e3·45-s − 97.2·47-s − 315.·49-s − 676.·51-s + 209.·53-s + 143.·55-s + ⋯
L(s)  = 1  − 1.90·3-s + 1.50·5-s + 0.281·7-s + 2.61·9-s + 0.234·11-s + 0.241·13-s − 2.86·15-s + 0.976·17-s − 0.0836·19-s − 0.535·21-s − 1.19·23-s + 1.26·25-s − 3.07·27-s + 0.185·29-s − 0.00242·31-s − 0.445·33-s + 0.424·35-s − 1.75·37-s − 0.460·39-s − 1.70·41-s − 0.653·43-s + 3.93·45-s − 0.301·47-s − 0.920·49-s − 1.85·51-s + 0.543·53-s + 0.352·55-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 \( 1 + 9.87T + 27T^{2} \)
5 \( 1 - 16.8T + 125T^{2} \)
7 \( 1 - 5.21T + 343T^{2} \)
11 \( 1 - 8.55T + 1.33e3T^{2} \)
13 \( 1 - 11.3T + 2.19e3T^{2} \)
17 \( 1 - 68.4T + 4.91e3T^{2} \)
19 \( 1 + 6.93T + 6.85e3T^{2} \)
23 \( 1 + 132.T + 1.21e4T^{2} \)
31 \( 1 + 0.419T + 2.97e4T^{2} \)
37 \( 1 + 395.T + 5.06e4T^{2} \)
41 \( 1 + 447.T + 6.89e4T^{2} \)
43 \( 1 + 184.T + 7.95e4T^{2} \)
47 \( 1 + 97.2T + 1.03e5T^{2} \)
53 \( 1 - 209.T + 1.48e5T^{2} \)
59 \( 1 + 45.9T + 2.05e5T^{2} \)
61 \( 1 + 427.T + 2.26e5T^{2} \)
67 \( 1 - 405.T + 3.00e5T^{2} \)
71 \( 1 + 557.T + 3.57e5T^{2} \)
73 \( 1 + 381.T + 3.89e5T^{2} \)
79 \( 1 - 577.T + 4.93e5T^{2} \)
83 \( 1 - 353.T + 5.71e5T^{2} \)
89 \( 1 + 277.T + 7.04e5T^{2} \)
97 \( 1 - 677.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.566540692704512765023375134665, −7.39451341743459973178056683761, −6.49484031661583710810281065130, −6.08591004408998976445317165166, −5.32049357261464651929409068216, −4.86496192334696858442598665179, −3.60369921260683542776789615104, −1.86844166743318844357118936195, −1.31372328702898687549741934268, 0, 1.31372328702898687549741934268, 1.86844166743318844357118936195, 3.60369921260683542776789615104, 4.86496192334696858442598665179, 5.32049357261464651929409068216, 6.08591004408998976445317165166, 6.49484031661583710810281065130, 7.39451341743459973178056683761, 8.566540692704512765023375134665

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