Properties

Label 2-2368-148.47-c0-0-0
Degree $2$
Conductor $2368$
Sign $-0.929 - 0.367i$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
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  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 2i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.499i)15-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.499 − 0.866i)21-s i·27-s − 2·29-s + (−1 − 1.73i)33-s + (−0.866 − 0.499i)35-s − 37-s + (−0.866 − 0.499i)39-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 2i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.499i)15-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.499 − 0.866i)21-s i·27-s − 2·29-s + (−1 − 1.73i)33-s + (−0.866 − 0.499i)35-s − 37-s + (−0.866 − 0.499i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $-0.929 - 0.367i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ -0.929 - 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7299111879\)
\(L(\frac12)\) \(\approx\) \(0.7299111879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + T \)
good3 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 - 2iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.563060583749866065976975407476, −9.227277990226727676719224399743, −7.70703894349182852654001089648, −6.96098326942010593682754343903, −6.46641193038376928670583112525, −5.52937689194565315160023299661, −4.93657487872530910180363832398, −3.93115770749142610021613306321, −2.80247311086365203012754978867, −1.93413472847654885448222898557, 0.57599081996046052600021127654, 1.42250530226885003740679661387, 3.32328286489725431539489230532, 3.66430968086689078933731541613, 5.42403904498155839031749882017, 5.67895843105368293830297591377, 6.18000697797627914759832250468, 7.20355267944855152147036874962, 8.088418882388279938125382683533, 8.864512977739521613915007403203

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