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Properties

Label 2-2736-228.83-c1-0-6
Degree $2$
Conductor $2736$
Sign $-0.836 + 0.548i$
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  + (−2.03 + 1.17i)5-s + 4.52i·7-s + 2.20·11-s + (−1.93 + 3.35i)13-s + (−5.29 + 3.05i)17-s + (−2.75 + 3.37i)19-s + (−0.557 + 0.966i)23-s + (0.270 − 0.468i)25-s + (−3.51 − 2.02i)29-s − 6.92i·31-s + (−5.33 − 9.23i)35-s + 10.9·37-s + (8.80 − 5.08i)41-s + (−9.76 + 5.63i)43-s + (6.43 − 11.1i)47-s + ⋯
L(s)  = 1  + (−0.911 + 0.526i)5-s + 1.71i·7-s + 0.666·11-s + (−0.536 + 0.929i)13-s + (−1.28 + 0.740i)17-s + (−0.632 + 0.774i)19-s + (−0.116 + 0.201i)23-s + (0.0541 − 0.0937i)25-s + (−0.652 − 0.376i)29-s − 1.24i·31-s + (−0.900 − 1.56i)35-s + 1.80·37-s + (1.37 − 0.794i)41-s + (−1.48 + 0.859i)43-s + (0.938 − 1.62i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.836 + 0.548i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.836 + 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5858700626\)
\(L(\frac12)\) \(\approx\) \(0.5858700626\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.75 - 3.37i)T \)
good5 \( 1 + (2.03 - 1.17i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.52iT - 7T^{2} \)
11 \( 1 - 2.20T + 11T^{2} \)
13 \( 1 + (1.93 - 3.35i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.29 - 3.05i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.557 - 0.966i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.51 + 2.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + (-8.80 + 5.08i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.76 - 5.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.43 + 11.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.32 - 4.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.66 + 6.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.719 - 1.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.28 - 3.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.87 - 6.70i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.705 - 1.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.92 - 1.11i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (-2.44 - 1.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.76 + 6.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.208127605616235055994991750886, −8.524641237030559175705403926597, −7.86979180564610252594516872359, −6.93139422126008973731339501052, −6.21734161929991262482714060780, −5.59432348729332657735849591763, −4.28744789924761661597466860779, −3.89723966199991858896610660644, −2.53376038832347652611238450726, −1.97021092906897771512538795476, 0.21584149206759962576963172130, 1.04695297542454454820227289136, 2.64828180964245436560684241648, 3.75951217586694011222254642945, 4.42284983743817971463980606742, 4.86491248541702288203540843138, 6.26869461675938467397204494906, 7.06541736722255383715232252087, 7.52819860969739415717185546273, 8.290784130561919081018860143093

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