Properties

Label 2-2394-21.20-c1-0-44
Degree $2$
Conductor $2394$
Sign $-0.900 - 0.434i$
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 + 1.62·5-s + (−2.31 + 1.28i)7-s + i·8-s − 1.62i·10-s + 0.0154i·11-s − 4.25i·13-s + (1.28 + 2.31i)14-s + 16-s − 3.03·17-s i·19-s − 1.62·20-s + 0.0154·22-s + 4.14i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.727·5-s + (−0.874 + 0.484i)7-s + 0.353i·8-s − 0.514i·10-s + 0.00466i·11-s − 1.17i·13-s + (0.342 + 0.618i)14-s + 0.250·16-s − 0.736·17-s − 0.229i·19-s − 0.363·20-s + 0.00329·22-s + 0.863i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -0.900 - 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3341555926\)
\(L(\frac12)\) \(\approx\) \(0.3341555926\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.31 - 1.28i)T \)
19 \( 1 + iT \)
good5 \( 1 - 1.62T + 5T^{2} \)
11 \( 1 - 0.0154iT - 11T^{2} \)
13 \( 1 + 4.25iT - 13T^{2} \)
17 \( 1 + 3.03T + 17T^{2} \)
23 \( 1 - 4.14iT - 23T^{2} \)
29 \( 1 + 5.72iT - 29T^{2} \)
31 \( 1 - 3.69iT - 31T^{2} \)
37 \( 1 - 4.31T + 37T^{2} \)
41 \( 1 + 4.11T + 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 + 5.85T + 47T^{2} \)
53 \( 1 + 3.96iT - 53T^{2} \)
59 \( 1 + 9.76T + 59T^{2} \)
61 \( 1 - 3.02iT - 61T^{2} \)
67 \( 1 + 8.17T + 67T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 0.624T + 83T^{2} \)
89 \( 1 + 9.38T + 89T^{2} \)
97 \( 1 - 1.70iT - 97T^{2} \)
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

−8.706348795048509220935258951911, −7.954471802262324766042553162995, −6.88808790866605847231827635307, −5.97250207473061031810216623113, −5.49006250037390892132341448439, −4.43075069760098752768562589711, −3.31044306264189529958455222559, −2.67522806048074982310894989463, −1.64210427655784113747970361810, −0.10674618443436288596812094295, 1.58501883278131080833025367657, 2.80055329850873464619854963834, 3.97650815167298715977333078954, 4.63978729720383877397154309058, 5.72990892711467200129398035851, 6.46089800754356763635623768831, 6.84077376838453516835404617342, 7.75510488884698802053418651137, 8.757909582535943086442774033609, 9.315873721022934647277379744681

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