↑

Properties

Label 2-2664-296.147-c0-0-9
Degree $2$
Conductor $2664$
Sign $-0.923 + 0.382i$
Analytic cond. $1.32950$
Root an. cond. $1.15304$
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  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s − 0.765·5-s − 1.41i·7-s + (−0.382 − 0.923i)8-s + (0.707 + 0.292i)10-s + (−0.541 + 1.30i)14-s + i·16-s − 0.765i·17-s + (−0.541 − 0.541i)20-s + 0.765·23-s − 0.414·25-s + (1.00 − i)28-s − 1.84·29-s + (0.382 − 0.923i)32-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s − 0.765·5-s − 1.41i·7-s + (−0.382 − 0.923i)8-s + (0.707 + 0.292i)10-s + (−0.541 + 1.30i)14-s + i·16-s − 0.765i·17-s + (−0.541 − 0.541i)20-s + 0.765·23-s − 0.414·25-s + (1.00 − i)28-s − 1.84·29-s + (0.382 − 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(1.32950\)
Root analytic conductor: \(1.15304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2664,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3983277879\)
\(L(\frac12)\) \(\approx\) \(0.3983277879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
3 \( 1 \)
37 \( 1 + iT \)
good5 \( 1 + 0.765T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 0.765iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 0.765T + T^{2} \)
29 \( 1 + 1.84T + T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 0.765iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.84iT - T^{2} \)
97 \( 1 - T^{2} \)
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

−8.813880145621067034141104759713, −7.74173705699010224570343995852, −7.43336657482870751956581663509, −6.90356441365518962552395577426, −5.72335723556042933521952395266, −4.39753638504606604987738396501, −3.79818255645462344592512164493, −2.95350266444204681305656460230, −1.59495025831135935076569858307, −0.35051348728392832652617393472, 1.60554284171153691975487844408, 2.59515814488071465684412498713, 3.64259185657015584920750699319, 4.92701168259458281436560953163, 5.71510188851245779474530181668, 6.33419632878224189316509463922, 7.31799296243971493500020565182, 7.941258978186715583313937411962, 8.643423653855809357065213851057, 9.145374038649773001488748856142

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