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Properties

Label 2-2736-57.8-c1-0-10
Degree $2$
Conductor $2736$
Sign $-0.349 - 0.936i$
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  + (3.15 + 1.82i)5-s − 1.21·7-s + 3.18i·11-s + (3.24 − 1.87i)13-s + (−5.49 − 3.17i)17-s + (−1.51 + 4.08i)19-s + (−8.02 + 4.63i)23-s + (4.15 + 7.20i)25-s + (2.76 + 4.79i)29-s + 7.28i·31-s + (−3.84 − 2.22i)35-s − 7.63i·37-s + (−5.22 + 9.04i)41-s + (2.58 − 4.47i)43-s + (9.64 − 5.56i)47-s + ⋯
L(s)  = 1  + (1.41 + 0.815i)5-s − 0.460·7-s + 0.961i·11-s + (0.899 − 0.519i)13-s + (−1.33 − 0.769i)17-s + (−0.348 + 0.937i)19-s + (−1.67 + 0.965i)23-s + (0.831 + 1.44i)25-s + (0.513 + 0.889i)29-s + 1.30i·31-s + (−0.650 − 0.375i)35-s − 1.25i·37-s + (−0.815 + 1.41i)41-s + (0.393 − 0.681i)43-s + (1.40 − 0.811i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.349 - 0.936i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.770212051\)
\(L(\frac12)\) \(\approx\) \(1.770212051\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 4.08i)T \)
good5 \( 1 + (-3.15 - 1.82i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.21T + 7T^{2} \)
11 \( 1 - 3.18iT - 11T^{2} \)
13 \( 1 + (-3.24 + 1.87i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.49 + 3.17i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (8.02 - 4.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.76 - 4.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.28iT - 31T^{2} \)
37 \( 1 + 7.63iT - 37T^{2} \)
41 \( 1 + (5.22 - 9.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.58 + 4.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.64 + 5.56i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.30 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.27 + 7.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.41 - 2.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.06 - 3.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.96 - 3.39i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.32 - 2.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.33 + 4.23i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.122iT - 83T^{2} \)
89 \( 1 + (-5.93 - 10.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.33 + 4.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.175668759530187100359018544859, −8.440130973089912822357809114866, −7.26940239418144798787001217701, −6.75995957399517994083184228576, −5.98404713554113500667035673705, −5.46346767095856741262194114820, −4.27446522136781534562114927074, −3.29021797076794249086875910830, −2.33379474727708620537117266939, −1.59435934969215217940005542971, 0.52643278678389901325589386421, 1.85388788294108145266865503652, 2.55215285229022159393596507481, 3.97209128154851204098818565509, 4.59034045052065287297350875059, 5.81325674492600791160826605781, 6.15733355764821827322658863853, 6.69740833699060184526403234029, 8.220950352203549254716603162055, 8.657705881627994353688052067724

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