↑

Properties

Label 2-2736-228.83-c1-0-22
Degree $2$
Conductor $2736$
Sign $0.194 + 0.980i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + (−2.15 + 1.24i)5-s − 0.872i·7-s − 2.41·11-s + (−1.98 + 3.43i)13-s + (0.303 − 0.174i)17-s + (1.35 + 4.14i)19-s + (3.88 − 6.72i)23-s + (0.608 − 1.05i)25-s + (−7.96 − 4.59i)29-s + 2.43i·31-s + (1.08 + 1.88i)35-s + 6.18·37-s + (−1.31 + 0.760i)41-s + (−5.24 + 3.02i)43-s + (4.05 − 7.01i)47-s + ⋯
L(s)  = 1  + (−0.965 + 0.557i)5-s − 0.329i·7-s − 0.726·11-s + (−0.549 + 0.951i)13-s + (0.0734 − 0.0424i)17-s + (0.311 + 0.950i)19-s + (0.809 − 1.40i)23-s + (0.121 − 0.210i)25-s + (−1.47 − 0.853i)29-s + 0.437i·31-s + (0.183 + 0.318i)35-s + 1.01·37-s + (−0.205 + 0.118i)41-s + (−0.800 + 0.462i)43-s + (0.590 − 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7152864871\)
\(L(\frac12)\) \(\approx\) \(0.7152864871\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-1.35 - 4.14i)T \)
good5 \( 1 + (2.15 - 1.24i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.872iT - 7T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
13 \( 1 + (1.98 - 3.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.303 + 0.174i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.88 + 6.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.96 + 4.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.43iT - 31T^{2} \)
37 \( 1 - 6.18T + 37T^{2} \)
41 \( 1 + (1.31 - 0.760i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.24 - 3.02i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.05 + 7.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.41 + 1.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.55 + 2.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.53 + 11.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.04 + 2.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.65 - 8.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.01 + 1.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.8 + 6.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.29T + 83T^{2} \)
89 \( 1 + (14.0 + 8.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.90 + 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.477310669691246422876907302721, −7.82253903827016013469954614095, −7.21293710583598718767621044758, −6.58126583205219081753025189765, −5.52722724288529507366607748991, −4.58598634171570503150836731555, −3.87404879465816493124939449057, −3.00969751127182378784977579642, −1.95806067323206694859156848102, −0.28312338557777281910389952844, 0.953372675643399093162194018699, 2.48771345240464280570803290430, 3.33064726997735073543869692226, 4.26993793582480577690244849761, 5.26754185623329174142406555050, 5.54721453284986752503540125295, 6.95551472176271749747910773146, 7.65903386862477660314225839735, 8.028853996573026832822380541108, 9.061123840458669096170338658757

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