Properties

Label 2-2394-57.56-c1-0-24
Degree $2$
Conductor $2394$
Sign $0.904 + 0.426i$
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  + 2-s + 4-s − 1.15i·5-s − 7-s + 8-s − 1.15i·10-s + 2.90i·11-s − 1.38i·13-s − 14-s + 16-s + 2.90i·17-s + (3.79 − 2.14i)19-s − 1.15i·20-s + 2.90i·22-s − 7.14i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.516i·5-s − 0.377·7-s + 0.353·8-s − 0.365i·10-s + 0.876i·11-s − 0.383i·13-s − 0.267·14-s + 0.250·16-s + 0.704i·17-s + (0.870 − 0.492i)19-s − 0.258i·20-s + 0.619i·22-s − 1.49i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.835588674\)
\(L(\frac12)\) \(\approx\) \(2.835588674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
19 \( 1 + (-3.79 + 2.14i)T \)
good5 \( 1 + 1.15iT - 5T^{2} \)
11 \( 1 - 2.90iT - 11T^{2} \)
13 \( 1 + 1.38iT - 13T^{2} \)
17 \( 1 - 2.90iT - 17T^{2} \)
23 \( 1 + 7.14iT - 23T^{2} \)
29 \( 1 - 4.66T + 29T^{2} \)
31 \( 1 - 4.06iT - 31T^{2} \)
37 \( 1 + 6.55iT - 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 2.04T + 43T^{2} \)
47 \( 1 - 2.76iT - 47T^{2} \)
53 \( 1 - 7.39T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.29T + 61T^{2} \)
67 \( 1 + 9.91iT - 67T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 - 14.5iT - 83T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + 3.92iT - 97T^{2} \)
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

−8.909661298522280591003362705532, −8.095117645073017626774218191251, −7.21383478895818524947975668195, −6.54860259133399355086203141675, −5.68983387172179462855958174807, −4.82721178114342681805022215331, −4.26866435678135092271891355540, −3.15752282720635513631025011471, −2.28363232779957446998454783609, −0.920840266840014272614298634841, 1.11637436356076120596049565984, 2.61183987231393619658795783646, 3.25145137255268590413411492910, 4.07796919582929217715903036978, 5.17685715011732312125880056534, 5.84327013359402077558600374632, 6.62905579628947495032969003137, 7.32121696175275359855389653386, 8.079749706229315404423197851205, 9.115600741773889567060854219895

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