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Properties

Label 2-2736-57.50-c1-0-7
Degree $2$
Conductor $2736$
Sign $-0.244 - 0.969i$
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  + (−1.11 + 0.646i)5-s + 0.567·7-s − 5.19i·11-s + (−6.03 − 3.48i)13-s + (−0.318 + 0.184i)17-s + (−1.16 + 4.20i)19-s + (6.57 + 3.79i)23-s + (−1.66 + 2.88i)25-s + (−3.70 + 6.41i)29-s − 6.51i·31-s + (−0.635 + 0.366i)35-s − 4.89i·37-s + (5.62 + 9.73i)41-s + (2.44 + 4.24i)43-s + (7.37 + 4.25i)47-s + ⋯
L(s)  = 1  + (−0.500 + 0.288i)5-s + 0.214·7-s − 1.56i·11-s + (−1.67 − 0.966i)13-s + (−0.0773 + 0.0446i)17-s + (−0.267 + 0.963i)19-s + (1.37 + 0.791i)23-s + (−0.332 + 0.576i)25-s + (−0.687 + 1.19i)29-s − 1.17i·31-s + (−0.107 + 0.0619i)35-s − 0.805i·37-s + (0.877 + 1.52i)41-s + (0.373 + 0.646i)43-s + (1.07 + 0.621i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.244 - 0.969i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.244 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7593580284\)
\(L(\frac12)\) \(\approx\) \(0.7593580284\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 - 4.20i)T \)
good5 \( 1 + (1.11 - 0.646i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.567T + 7T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + (6.03 + 3.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.318 - 0.184i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.57 - 3.79i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.70 - 6.41i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.51iT - 31T^{2} \)
37 \( 1 + 4.89iT - 37T^{2} \)
41 \( 1 + (-5.62 - 9.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.44 - 4.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.37 - 4.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.03 - 5.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.03 - 5.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.52 + 2.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.10 + 0.640i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.67 + 6.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.43 - 12.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.12 - 4.69i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + (0.953 - 1.65i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.75 + 2.16i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.135424408396709145819317550348, −7.950884312480452211776133325945, −7.78145055549165825015228905358, −6.90193033357012301698370013624, −5.74998185388071841301323769564, −5.37521938230107575414000484134, −4.22978392830546825382577558896, −3.29457315680552296969557108664, −2.66388495976463400960211820385, −1.11636709968838903457616014398, 0.26621632695371968494976680659, 1.96245001300080832119552879976, 2.60886023315566659350866508433, 4.10436382546422903642330253543, 4.67158183269506014229614952716, 5.15654827937318197772485680734, 6.62126458486557548089643531685, 7.13738152032799709977739964906, 7.65818606747409463415235599066, 8.732905081137333103909855609097

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