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Properties

Label 2-2736-76.31-c1-0-38
Degree $2$
Conductor $2736$
Sign $-0.211 + 0.977i$
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  + (0.876 + 1.51i)5-s + i·7-s + 1.75i·11-s + (−1.5 − 0.866i)13-s + (−2.39 − 4.14i)17-s + (−1.73 − 4i)19-s + (−5.66 − 3.27i)23-s + (0.964 − 1.66i)25-s + (−7.18 − 4.14i)29-s − 7.73·31-s + (−1.51 + 0.876i)35-s − 7.73i·37-s + (0.401 − 0.232i)43-s + (8.29 + 4.78i)47-s + 6·49-s + ⋯
L(s)  = 1  + (0.391 + 0.678i)5-s + 0.377i·7-s + 0.528i·11-s + (−0.416 − 0.240i)13-s + (−0.580 − 1.00i)17-s + (−0.397 − 0.917i)19-s + (−1.18 − 0.681i)23-s + (0.192 − 0.333i)25-s + (−1.33 − 0.770i)29-s − 1.38·31-s + (−0.256 + 0.148i)35-s − 1.27i·37-s + (0.0612 − 0.0353i)43-s + (1.20 + 0.698i)47-s + 0.857·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.211 + 0.977i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.211 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8478334961\)
\(L(\frac12)\) \(\approx\) \(0.8478334961\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 4i)T \)
good5 \( 1 + (-0.876 - 1.51i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 1.75iT - 11T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.39 + 4.14i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.66 + 3.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.18 + 4.14i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.73T + 31T^{2} \)
37 \( 1 + 7.73iT - 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.401 + 0.232i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.29 - 4.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.81 - 5.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.66 - 9.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.96 + 8.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.401 - 0.696i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.18 + 12.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.69 + 13.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.33 + 2.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.28iT - 83T^{2} \)
89 \( 1 + (-4.55 - 2.62i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.936370959180960570167304118637, −7.51396523769647505600501180635, −7.28826687611775937016392490666, −6.26262492366969467849514561974, −5.63713480076676472937120067294, −4.66705065192649561921341587507, −3.84295683955887230256869664417, −2.44154771035472330406969209332, −2.27009995490689494735203610375, −0.25891975739510022686354904376, 1.36828566199208664433434380355, 2.17317558485577016707608242194, 3.69137320302823083045817405966, 4.10496642822290018776566331793, 5.44403320835014132505885750977, 5.69176023227666542320837877343, 6.81491491322129657079568481421, 7.51571946493871493557648134482, 8.492581840307745846540234540395, 8.860592435406699812455645183030

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