Properties

Label 2-1856-1.1-c3-0-139
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  + 9.74·3-s − 17.5·5-s − 10.7·7-s + 67.8·9-s + 5.60·11-s − 17.1·13-s − 170.·15-s − 100.·17-s + 76.5·19-s − 104.·21-s + 76.5·23-s + 182.·25-s + 398.·27-s + 29·29-s + 181.·31-s + 54.5·33-s + 188.·35-s + 175.·37-s − 166.·39-s + 153.·41-s − 441.·43-s − 1.19e3·45-s − 431.·47-s − 227.·49-s − 977.·51-s + 178.·53-s − 98.3·55-s + ⋯
L(s)  = 1  + 1.87·3-s − 1.56·5-s − 0.579·7-s + 2.51·9-s + 0.153·11-s − 0.364·13-s − 2.94·15-s − 1.43·17-s + 0.923·19-s − 1.08·21-s + 0.694·23-s + 1.46·25-s + 2.83·27-s + 0.185·29-s + 1.04·31-s + 0.287·33-s + 0.910·35-s + 0.780·37-s − 0.684·39-s + 0.586·41-s − 1.56·43-s − 3.94·45-s − 1.33·47-s − 0.663·49-s − 2.68·51-s + 0.462·53-s − 0.241·55-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 \( 1 - 9.74T + 27T^{2} \)
5 \( 1 + 17.5T + 125T^{2} \)
7 \( 1 + 10.7T + 343T^{2} \)
11 \( 1 - 5.60T + 1.33e3T^{2} \)
13 \( 1 + 17.1T + 2.19e3T^{2} \)
17 \( 1 + 100.T + 4.91e3T^{2} \)
19 \( 1 - 76.5T + 6.85e3T^{2} \)
23 \( 1 - 76.5T + 1.21e4T^{2} \)
31 \( 1 - 181.T + 2.97e4T^{2} \)
37 \( 1 - 175.T + 5.06e4T^{2} \)
41 \( 1 - 153.T + 6.89e4T^{2} \)
43 \( 1 + 441.T + 7.95e4T^{2} \)
47 \( 1 + 431.T + 1.03e5T^{2} \)
53 \( 1 - 178.T + 1.48e5T^{2} \)
59 \( 1 + 765.T + 2.05e5T^{2} \)
61 \( 1 + 611.T + 2.26e5T^{2} \)
67 \( 1 + 729.T + 3.00e5T^{2} \)
71 \( 1 + 552.T + 3.57e5T^{2} \)
73 \( 1 + 753.T + 3.89e5T^{2} \)
79 \( 1 + 917.T + 4.93e5T^{2} \)
83 \( 1 - 225.T + 5.71e5T^{2} \)
89 \( 1 - 86.6T + 7.04e5T^{2} \)
97 \( 1 + 1.67e3T + 9.12e5T^{2} \)
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

−8.428066444271966622823681662280, −7.83742606651996775966762446949, −7.20237587827492080033620163324, −6.54009265649579301822220097468, −4.63683039190104491527944744914, −4.24502392065890446520270319642, −3.10803653125865008567545667265, −2.93624279897876652459736183688, −1.45900456278828047682619166542, 0, 1.45900456278828047682619166542, 2.93624279897876652459736183688, 3.10803653125865008567545667265, 4.24502392065890446520270319642, 4.63683039190104491527944744914, 6.54009265649579301822220097468, 7.20237587827492080033620163324, 7.83742606651996775966762446949, 8.428066444271966622823681662280

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