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Properties

Label 2-2736-57.8-c1-0-38
Degree $2$
Conductor $2736$
Sign $-0.900 - 0.435i$
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  + (−3.20 − 1.85i)5-s + 4.57·7-s − 0.467i·11-s + (−1.80 + 1.04i)13-s + (−1.39 − 0.806i)17-s + (−4.34 + 0.294i)19-s + (−3.86 + 2.22i)23-s + (4.34 + 7.53i)25-s + (−3.12 − 5.41i)29-s − 5.77i·31-s + (−14.6 − 8.45i)35-s + 8.44i·37-s + (2.87 − 4.98i)41-s + (2.14 − 3.70i)43-s + (−6.43 + 3.71i)47-s + ⋯
L(s)  = 1  + (−1.43 − 0.827i)5-s + 1.72·7-s − 0.141i·11-s + (−0.501 + 0.289i)13-s + (−0.338 − 0.195i)17-s + (−0.997 + 0.0674i)19-s + (−0.805 + 0.464i)23-s + (0.869 + 1.50i)25-s + (−0.580 − 1.00i)29-s − 1.03i·31-s + (−2.47 − 1.42i)35-s + 1.38i·37-s + (0.449 − 0.777i)41-s + (0.326 − 0.565i)43-s + (−0.939 + 0.542i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.900 - 0.435i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04452019499\)
\(L(\frac12)\) \(\approx\) \(0.04452019499\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.34 - 0.294i)T \)
good5 \( 1 + (3.20 + 1.85i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.57T + 7T^{2} \)
11 \( 1 + 0.467iT - 11T^{2} \)
13 \( 1 + (1.80 - 1.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.39 + 0.806i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.86 - 2.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.12 + 5.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.77iT - 31T^{2} \)
37 \( 1 - 8.44iT - 37T^{2} \)
41 \( 1 + (-2.87 + 4.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.14 + 3.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.43 - 3.71i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.22 - 9.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.70 + 2.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.52 + 7.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.15 - 5.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.31 - 5.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.69 - 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.02 + 1.17i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.13iT - 83T^{2} \)
89 \( 1 + (-4.20 - 7.27i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.11 + 1.22i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.288403935300527037009320201121, −7.79586379155686298769805726309, −7.23207051646973135648666732352, −5.94401515084227835303752045179, −5.03127775747086274423790933979, −4.30692399804938461361606325259, −4.01951186543976162011094358313, −2.41154111652792051637632147559, −1.36299512287462077501656508867, −0.01461618345918042705510746119, 1.68746871963635292457361626289, 2.70899005734513960025391395109, 3.84099826511252220183575779255, 4.45541956994498549486632474406, 5.17257159251876139115393060414, 6.35102830137117534432630005960, 7.26562998811364151821050547068, 7.70004467858020049135352593284, 8.349869467699398977940126938818, 8.939672778500591000441920837547

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