↑

Properties

Label 2-2736-76.31-c1-0-49
Degree $2$
Conductor $2736$
Sign $-0.659 - 0.751i$
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  + (−1.33 − 2.31i)5-s − 3.93i·7-s + 2.20i·11-s + (−3.60 − 2.07i)13-s + (0.571 + 0.990i)17-s + (−4.24 − 0.990i)19-s + (−3.19 − 1.84i)23-s + (−1.07 + 1.85i)25-s + (2.10 + 1.21i)29-s + 3.67·31-s + (−9.11 + 5.25i)35-s − 10.0i·37-s + (−8.01 + 4.62i)41-s + (−0.490 + 0.283i)43-s + (10.6 + 6.13i)47-s + ⋯
L(s)  = 1  + (−0.597 − 1.03i)5-s − 1.48i·7-s + 0.664i·11-s + (−0.998 − 0.576i)13-s + (0.138 + 0.240i)17-s + (−0.973 − 0.227i)19-s + (−0.665 − 0.384i)23-s + (−0.214 + 0.371i)25-s + (0.390 + 0.225i)29-s + 0.659·31-s + (−1.53 + 0.889i)35-s − 1.65i·37-s + (−1.25 + 0.723i)41-s + (−0.0748 + 0.0432i)43-s + (1.55 + 0.895i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3245949102\)
\(L(\frac12)\) \(\approx\) \(0.3245949102\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.24 + 0.990i)T \)
good5 \( 1 + (1.33 + 2.31i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.93iT - 7T^{2} \)
11 \( 1 - 2.20iT - 11T^{2} \)
13 \( 1 + (3.60 + 2.07i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.571 - 0.990i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.19 + 1.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.10 - 1.21i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.67T + 31T^{2} \)
37 \( 1 + 10.0iT - 37T^{2} \)
41 \( 1 + (8.01 - 4.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.490 - 0.283i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.6 - 6.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.09 + 2.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.33 - 2.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.41 - 11.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 + 3.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.24 - 10.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.35 - 2.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.50 + 6.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.98iT - 83T^{2} \)
89 \( 1 + (12.9 + 7.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.91 + 1.68i)T + (48.5 - 84.0i)T^{2} \)
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

−8.271194178949169613242357295585, −7.56671339181505091901446265785, −7.08438595805300849670679730585, −6.05734840585122605060114220358, −4.87184100134077535714011002600, −4.44206956973426094927649550065, −3.78420691725536498431461479310, −2.44145995979417562704916976541, −1.11335211582191243568453988943, −0.11304189279384620824298217223, 2.00563860365815473217296616193, 2.76083049711923952904135286663, 3.53664552375241879507488428782, 4.64300864439135456158848312423, 5.51424110527561760730537980899, 6.37098770431608098996524586919, 6.89072553832300704829028480012, 7.921632469426460224985709784840, 8.449671656573502996824975318424, 9.257053790799120260198837649158

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