↑

Properties

Label 2-2394-57.56-c1-0-13
Degree $2$
Conductor $2394$
Sign $0.662 - 0.749i$
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 − 2.82i·5-s + 7-s + 8-s − 2.82i·10-s + 5.65i·11-s + 4.24i·13-s + 14-s + 16-s + 5.65i·17-s + (1 + 4.24i)19-s − 2.82i·20-s + 5.65i·22-s + 5.65i·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.26i·5-s + 0.377·7-s + 0.353·8-s − 0.894i·10-s + 1.70i·11-s + 1.17i·13-s + 0.267·14-s + 0.250·16-s + 1.37i·17-s + (0.229 + 0.973i)19-s − 0.632i·20-s + 1.20i·22-s + 1.17i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.662 - 0.749i$
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: \((2,\ 2394,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.706913856\)
\(L(\frac12)\) \(\approx\) \(2.706913856\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 4.24i)T \)
good5 \( 1 + 2.82iT - 5T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 9.89iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 15.5iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 4.24iT - 97T^{2} \)
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

−9.310991589446234860422366920589, −8.056354211308018424464295117189, −7.66820708550829568687454781948, −6.63895417866900780216726933250, −5.75815983884809830930888930147, −4.99799762336026920409978685410, −4.28389351448283404441804741203, −3.77354349468577241075822986297, −1.91808674471370596432106588746, −1.63089110955943484759996039312, 0.70568919529652365780948334576, 2.55106370186651671794986832073, 3.02254564561446940417890854414, 3.77659001074163303563009639125, 5.17928792245381336342164630578, 5.50539109162830000917406999098, 6.74464452211536272954210434498, 6.93530347986999654208704314245, 8.060354390114408644113890028233, 8.655783755581797845159182363056

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