Properties

Label 2-2736-19.11-c1-0-20
Degree $2$
Conductor $2736$
Sign $0.827 - 0.560i$
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  + (0.795 − 1.37i)5-s + 3.87·7-s − 0.409·11-s + (1.64 + 2.84i)13-s + (−2.87 + 4.98i)17-s + (3.43 + 2.68i)19-s + (−0.0214 − 0.0372i)23-s + (1.23 + 2.14i)25-s + (2.69 + 4.66i)29-s − 0.773·31-s + (3.08 − 5.34i)35-s − 0.547·37-s + (−5.57 + 9.65i)41-s + (−3.26 + 5.65i)43-s + (−3.28 − 5.69i)47-s + ⋯
L(s)  = 1  + (0.355 − 0.615i)5-s + 1.46·7-s − 0.123·11-s + (0.455 + 0.788i)13-s + (−0.698 + 1.20i)17-s + (0.787 + 0.616i)19-s + (−0.00448 − 0.00776i)23-s + (0.247 + 0.428i)25-s + (0.500 + 0.866i)29-s − 0.138·31-s + (0.521 − 0.902i)35-s − 0.0899·37-s + (−0.870 + 1.50i)41-s + (−0.497 + 0.862i)43-s + (−0.479 − 0.830i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.827 - 0.560i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.827 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.327791986\)
\(L(\frac12)\) \(\approx\) \(2.327791986\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.43 - 2.68i)T \)
good5 \( 1 + (-0.795 + 1.37i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.87T + 7T^{2} \)
11 \( 1 + 0.409T + 11T^{2} \)
13 \( 1 + (-1.64 - 2.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.87 - 4.98i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.0214 + 0.0372i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.69 - 4.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.773T + 31T^{2} \)
37 \( 1 + 0.547T + 37T^{2} \)
41 \( 1 + (5.57 - 9.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.26 - 5.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.28 + 5.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.49 - 7.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.899 + 1.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.537 + 0.931i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.34 - 2.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.16 + 12.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.68 - 2.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + (5.85 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.02 + 13.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.616981330028726661912191331258, −8.406381102584916537982254227097, −7.50035501481164565900496488596, −6.56931487886678556259567548478, −5.71681741273755802357080548353, −4.90852038763915007655483657259, −4.38388572537942743141282457872, −3.28087435139670820330065105955, −1.75815082397422960287947272451, −1.42358871033553779482335397162, 0.801456732664297469397192462130, 2.12229064857856884187875991531, 2.85105754089066710719337385352, 4.01696661518682545568371202885, 5.05986050085656135259491116942, 5.41627535298783812730199629970, 6.61793629394379333140749391564, 7.20111628715197333310922922029, 8.068070214213355652312048381520, 8.600619131753174265474312415836

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