Properties

Label 2-1856-1.1-c3-0-72
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·3-s + 10.5·5-s + 13.7·7-s − 24.3·9-s + 30.2·11-s − 19.9·13-s − 16.9·15-s + 105.·17-s + 79.5·19-s − 22.1·21-s − 14.8·23-s − 14.0·25-s + 82.9·27-s − 29·29-s − 61.9·31-s − 48.8·33-s + 144.·35-s − 6.43·37-s + 32.1·39-s + 351.·41-s − 473.·43-s − 256.·45-s + 492.·47-s − 154.·49-s − 170.·51-s + 668.·53-s + 319.·55-s + ⋯
L(s)  = 1  − 0.310·3-s + 0.942·5-s + 0.741·7-s − 0.903·9-s + 0.830·11-s − 0.425·13-s − 0.292·15-s + 1.50·17-s + 0.960·19-s − 0.230·21-s − 0.134·23-s − 0.112·25-s + 0.591·27-s − 0.185·29-s − 0.358·31-s − 0.257·33-s + 0.698·35-s − 0.0285·37-s + 0.132·39-s + 1.33·41-s − 1.67·43-s − 0.851·45-s + 1.52·47-s − 0.449·49-s − 0.467·51-s + 1.73·53-s + 0.782·55-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: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.789334569\)
\(L(\frac12)\) \(\approx\) \(2.789334569\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.61T + 27T^{2} \)
5 \( 1 - 10.5T + 125T^{2} \)
7 \( 1 - 13.7T + 343T^{2} \)
11 \( 1 - 30.2T + 1.33e3T^{2} \)
13 \( 1 + 19.9T + 2.19e3T^{2} \)
17 \( 1 - 105.T + 4.91e3T^{2} \)
19 \( 1 - 79.5T + 6.85e3T^{2} \)
23 \( 1 + 14.8T + 1.21e4T^{2} \)
31 \( 1 + 61.9T + 2.97e4T^{2} \)
37 \( 1 + 6.43T + 5.06e4T^{2} \)
41 \( 1 - 351.T + 6.89e4T^{2} \)
43 \( 1 + 473.T + 7.95e4T^{2} \)
47 \( 1 - 492.T + 1.03e5T^{2} \)
53 \( 1 - 668.T + 1.48e5T^{2} \)
59 \( 1 + 12.7T + 2.05e5T^{2} \)
61 \( 1 + 80.8T + 2.26e5T^{2} \)
67 \( 1 + 529.T + 3.00e5T^{2} \)
71 \( 1 + 211.T + 3.57e5T^{2} \)
73 \( 1 - 627.T + 3.89e5T^{2} \)
79 \( 1 - 915.T + 4.93e5T^{2} \)
83 \( 1 + 411.T + 5.71e5T^{2} \)
89 \( 1 - 1.37e3T + 7.04e5T^{2} \)
97 \( 1 + 649.T + 9.12e5T^{2} \)
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

−9.020404463800279731506718531307, −8.063704705204508760238993998909, −7.35926621455780660907844684068, −6.30623115255850652551200504522, −5.55486923826456731393456313212, −5.16411820057230947720006156766, −3.88420325100670890257028654419, −2.83872342743203284801242329786, −1.76654340347678687381151072374, −0.831663111702098136619886812972, 0.831663111702098136619886812972, 1.76654340347678687381151072374, 2.83872342743203284801242329786, 3.88420325100670890257028654419, 5.16411820057230947720006156766, 5.55486923826456731393456313212, 6.30623115255850652551200504522, 7.35926621455780660907844684068, 8.063704705204508760238993998909, 9.020404463800279731506718531307

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