↑

Properties

Label 2-2793-2793.1466-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.925 + 0.377i$
Analytic cond. $1.39388$
Root an. cond. $1.18063$
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  + (−0.921 − 0.388i)3-s + (0.853 − 0.521i)4-s + (−0.411 + 0.911i)7-s + (0.698 + 0.715i)9-s + (−0.988 + 0.149i)12-s + (−1.58 + 0.158i)13-s + (0.456 − 0.889i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (0.969 − 0.246i)25-s + (−0.365 − 0.930i)27-s + (0.124 + 0.992i)28-s + 1.75·31-s + (0.969 + 0.246i)36-s + (0.970 + 0.381i)37-s + ⋯
L(s)  = 1  + (−0.921 − 0.388i)3-s + (0.853 − 0.521i)4-s + (−0.411 + 0.911i)7-s + (0.698 + 0.715i)9-s + (−0.988 + 0.149i)12-s + (−1.58 + 0.158i)13-s + (0.456 − 0.889i)16-s + (0.900 − 0.433i)19-s + (0.733 − 0.680i)21-s + (0.969 − 0.246i)25-s + (−0.365 − 0.930i)27-s + (0.124 + 0.992i)28-s + 1.75·31-s + (0.969 + 0.246i)36-s + (0.970 + 0.381i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.925 + 0.377i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1466, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.051937042\)
\(L(\frac12)\) \(\approx\) \(1.051937042\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.921 + 0.388i)T \)
7 \( 1 + (0.411 - 0.911i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (-0.853 + 0.521i)T^{2} \)
5 \( 1 + (-0.969 + 0.246i)T^{2} \)
11 \( 1 + (-0.365 - 0.930i)T^{2} \)
13 \( 1 + (1.58 - 0.158i)T + (0.980 - 0.198i)T^{2} \)
17 \( 1 + (0.456 + 0.889i)T^{2} \)
23 \( 1 + (0.998 - 0.0498i)T^{2} \)
29 \( 1 + (0.921 + 0.388i)T^{2} \)
31 \( 1 - 1.75T + T^{2} \)
37 \( 1 + (-0.970 - 0.381i)T + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (0.969 - 0.246i)T^{2} \)
43 \( 1 + (-1.39 + 0.0696i)T + (0.995 - 0.0995i)T^{2} \)
47 \( 1 + (0.980 - 0.198i)T^{2} \)
53 \( 1 + (0.456 - 0.889i)T^{2} \)
59 \( 1 + (0.411 - 0.911i)T^{2} \)
61 \( 1 + (0.189 - 0.937i)T + (-0.921 - 0.388i)T^{2} \)
67 \( 1 + (1.18 - 0.209i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.124 + 0.992i)T^{2} \)
73 \( 1 + (-1.62 - 1.16i)T + (0.318 + 0.947i)T^{2} \)
79 \( 1 + (-0.963 + 1.14i)T + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.988 + 0.149i)T^{2} \)
89 \( 1 + (0.853 + 0.521i)T^{2} \)
97 \( 1 + (0.114 + 0.0960i)T + (0.173 + 0.984i)T^{2} \)
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

−9.125158561432769666733925478837, −7.900226896964877874900785755013, −7.24283564483915142424459455778, −6.58073359077743183897772369889, −5.97056008372167601490164906940, −5.18954046465715521443659007248, −4.61528293933584479464200052446, −2.82092545140302331847217827763, −2.36023830339938739383845337704, −0.991071005433459335605816176962, 1.00773125097396495926518628745, 2.55962350837069015070621559795, 3.42916068251288897634729050093, 4.37153878324234146325624660284, 5.10329812210740629423812278359, 6.13937545410443907870006537183, 6.74449755078463292104080797569, 7.44666148392301434192534847548, 7.911694138239223052040029498025, 9.335603100401742517174566807977

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