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Properties

Label 2-2736-76.27-c1-0-4
Degree $2$
Conductor $2736$
Sign $-0.977 + 0.211i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)5-s + 1.09i·7-s + 5.65i·11-s + (−0.949 + 0.548i)13-s + (−1 + 1.73i)17-s + (−4 − 1.73i)19-s + (4.89 − 2.82i)23-s + (0.500 + 0.866i)25-s + (−7.89 + 4.56i)29-s + 1.89·31-s + (−1.89 − 1.09i)35-s − 4.56i·37-s + (−4.5 − 2.59i)43-s + (−1.89 + 1.09i)47-s + 5.79·49-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + 0.414i·7-s + 1.70i·11-s + (−0.263 + 0.152i)13-s + (−0.242 + 0.420i)17-s + (−0.917 − 0.397i)19-s + (1.02 − 0.589i)23-s + (0.100 + 0.173i)25-s + (−1.46 + 0.846i)29-s + 0.341·31-s + (−0.320 − 0.185i)35-s − 0.749i·37-s + (−0.686 − 0.396i)43-s + (−0.276 + 0.159i)47-s + 0.828·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6461960958\)
\(L(\frac12)\) \(\approx\) \(0.6461960958\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.09iT - 7T^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (0.949 - 0.548i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.89 + 2.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.89 - 4.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.89T + 31T^{2} \)
37 \( 1 + 4.56iT - 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.89 - 1.09i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.10 - 0.635i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.89 + 6.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0505 - 0.0874i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.39 + 9.35i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2 + 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.39 - 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.94 + 8.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (7.89 - 4.56i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.7 + 9.12i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.195564191586150372398494711274, −8.548749255115015572096002580185, −7.44095517250264847479499747098, −7.08141905192414319923591143012, −6.38969440184186089426056016233, −5.23990304417761435628456587380, −4.52529785701122290258898890437, −3.64809597811468439805935928182, −2.57935590553085286662923100501, −1.81088852192698189906010984810, 0.21674147845568428591423726115, 1.23313211643731647725587168954, 2.71584093443739425912032741263, 3.66089573274144762578805604312, 4.40717776293095458109390521273, 5.32075995191703536650629304612, 6.02793302061106310308813293156, 6.94734619145234452012663613193, 7.82296954940217713795667966292, 8.463006312698456071439425481560

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