↑

Properties

Label 2-2394-133.132-c1-0-34
Degree $2$
Conductor $2394$
Sign $0.633 - 0.773i$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + 0.575i·5-s + (−2.19 + 1.47i)7-s − i·8-s − 0.575·10-s + 5.11·11-s + 6.37·13-s + (−1.47 − 2.19i)14-s + 16-s − 1.63i·17-s + (−1.26 − 4.17i)19-s − 0.575i·20-s + 5.11i·22-s − 6.12·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.257i·5-s + (−0.831 + 0.556i)7-s − 0.353i·8-s − 0.182·10-s + 1.54·11-s + 1.76·13-s + (−0.393 − 0.587i)14-s + 0.250·16-s − 0.395i·17-s + (−0.290 − 0.956i)19-s − 0.128i·20-s + 1.08i·22-s − 1.27·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.633 - 0.773i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.776252856\)
\(L(\frac12)\) \(\approx\) \(1.776252856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (2.19 - 1.47i)T \)
19 \( 1 + (1.26 + 4.17i)T \)
good5 \( 1 - 0.575iT - 5T^{2} \)
11 \( 1 - 5.11T + 11T^{2} \)
13 \( 1 - 6.37T + 13T^{2} \)
17 \( 1 + 1.63iT - 17T^{2} \)
23 \( 1 + 6.12T + 23T^{2} \)
29 \( 1 + 4.39iT - 29T^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + 7.51iT - 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 - 1.27T + 43T^{2} \)
47 \( 1 + 4.73iT - 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 14.8iT - 61T^{2} \)
67 \( 1 - 7.14iT - 67T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + 6.47iT - 73T^{2} \)
79 \( 1 - 7.82iT - 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 8.35T + 97T^{2} \)
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

−9.064765302159973417507574854280, −8.451153590866711536706463357242, −7.38988568124266716565928311152, −6.46904191400670721876084852105, −6.28753157814125525180637346660, −5.40711351057807243104883687751, −4.05404258996443546005213683330, −3.67384205833421634458248655999, −2.36467392409377748234509181751, −0.844487658583338636410177104120, 0.966819737810407555714137363562, 1.75210600644102062833258531601, 3.41099804520505992763032715821, 3.73293598505249961947679450051, 4.52822256973294339676023723971, 6.02577645843733037346128529647, 6.25062928621049689119563225604, 7.30916801756330059262676552691, 8.493147575183132174033001147081, 8.806622164760739249944605198401

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