↑

Properties

Label 2-2856-357.254-c0-0-3
Degree $2$
Conductor $2856$
Sign $0.319 + 0.947i$
Analytic cond. $1.42532$
Root an. cond. $1.19387$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (0.707 − 1.22i)5-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (0.707 + 1.22i)11-s + 13-s − 1.41i·15-s + (0.965 + 0.258i)17-s + (0.5 − 0.866i)19-s − 0.999·21-s + (−0.707 + 1.22i)23-s + (−0.499 − 0.866i)25-s − 0.999i·27-s − 1.41·29-s + (−0.866 + 0.5i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.707 − 1.22i)5-s + (−0.866 − 0.5i)7-s + (0.499 − 0.866i)9-s + (0.707 + 1.22i)11-s + 13-s − 1.41i·15-s + (0.965 + 0.258i)17-s + (0.5 − 0.866i)19-s − 0.999·21-s + (−0.707 + 1.22i)23-s + (−0.499 − 0.866i)25-s − 0.999i·27-s − 1.41·29-s + (−0.866 + 0.5i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2856\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.319 + 0.947i$
Analytic conductor: \(1.42532\)
Root analytic conductor: \(1.19387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2856} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2856,\ (\ :0),\ 0.319 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.885875729\)
\(L(\frac12)\) \(\approx\) \(1.885875729\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.965 - 0.258i)T \)
good5 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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

−9.103735688433698619392643311763, −7.992814831860587457953642879959, −7.44797044935043042293960841400, −6.56798013572538800720314139388, −5.87905872579523512819166136310, −4.88126563042608490795206250363, −3.87782610492449320486302101680, −3.27377037414567176684744207613, −1.77775738317802428645452345990, −1.28320853394255197714523170300, 1.73502031277520791382498677218, 2.82099563799014491427581784068, 3.39878198789800456992711796264, 3.94240178114751674911090134193, 5.62137032435037570225507693084, 5.99277383475269521173671405836, 6.73215741533396115879124650733, 7.72637909582986875128302442962, 8.442526307538294045663257874225, 9.304590124782993330693067369860

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