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Properties

Label 2-2736-76.27-c1-0-2
Degree $2$
Conductor $2736$
Sign $-0.740 + 0.671i$
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  + (−2.05 + 3.56i)5-s − i·7-s + 4.11i·11-s + (−1.5 + 0.866i)13-s + (−1.50 + 2.60i)17-s + (−1.73 + 4i)19-s + (0.954 − 0.551i)23-s + (−5.96 − 10.3i)25-s + (−4.51 + 2.60i)29-s + 4.26·31-s + (3.56 + 2.05i)35-s − 4.26i·37-s + (−5.59 − 3.23i)43-s + (5.21 − 3.01i)47-s + 6·49-s + ⋯
L(s)  = 1  + (−0.920 + 1.59i)5-s − 0.377i·7-s + 1.24i·11-s + (−0.416 + 0.240i)13-s + (−0.365 + 0.632i)17-s + (−0.397 + 0.917i)19-s + (0.199 − 0.114i)23-s + (−1.19 − 2.06i)25-s + (−0.838 + 0.484i)29-s + 0.766·31-s + (0.602 + 0.347i)35-s − 0.701i·37-s + (−0.853 − 0.492i)43-s + (0.760 − 0.439i)47-s + 0.857·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.740 + 0.671i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4287310190\)
\(L(\frac12)\) \(\approx\) \(0.4287310190\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 - 3.56i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 4.11iT - 11T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.50 - 2.60i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.954 + 0.551i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.51 - 2.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.26T + 31T^{2} \)
37 \( 1 + 4.26iT - 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.59 + 3.23i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.21 + 3.01i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.65 - 0.954i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.954 - 1.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.96 + 3.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.59 - 9.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.51 + 7.82i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.69 + 4.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.33 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 + (-10.6 + 6.17i)T + (44.5 - 77.0i)T^{2} \)
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

−9.428550251616977164091978874288, −8.282881027084625898923357248810, −7.65873394981260187989903178691, −6.94574413528262011078951774147, −6.64119239708902344689832494869, −5.47639496814222385605199173745, −4.19497657805888160871375849613, −3.89216159890499764276953263818, −2.76665942231057952209092302400, −1.88894658256382105155266900655, 0.15614822070705414483605247351, 1.07118789338785900002740522456, 2.56860293071202828999597214576, 3.60351297713783029901035087925, 4.53512416188518130874428114611, 5.09997527515487727374705504535, 5.87020396297735475735092857177, 6.91145598658896431683883323061, 7.912730789681181410401224678410, 8.321878342421972822098264552367

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