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Properties

Label 2-2280-2280.1019-c0-0-2
Degree $2$
Conductor $2280$
Sign $-0.582 + 0.813i$
Analytic cond. $1.13786$
Root an. cond. $1.06670$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999·10-s + (−0.866 − 0.499i)12-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + (−0.866 + 0.499i)18-s + (−0.5 − 0.866i)19-s + (0.866 + 0.499i)20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + 0.999·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s − 0.999·10-s + (−0.866 − 0.499i)12-s + (−0.499 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s + (−0.866 + 0.499i)18-s + (−0.5 − 0.866i)19-s + (0.866 + 0.499i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.582 + 0.813i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ -0.582 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4385765222\)
\(L(\frac12)\) \(\approx\) \(0.4385765222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.302134231497134795045542596163, −8.489372371023483302818291525857, −7.38242635935274160224688200692, −6.56222195710315610346723321764, −5.94550263447709024774483588216, −4.80415369779446560094778212880, −4.23147360148348476275492269912, −2.83103000977866775778828528583, −1.86082495664824207388427668846, −0.42107034883101781765897788991, 1.69614120498622067077696646814, 2.06286263321466900389891292489, 3.86076126554646135307943579802, 5.18021362259896222464727290424, 5.98203917410106343785344125743, 6.25774733151626002877519464436, 7.07074505029389312970365652991, 7.901563131776355809926065744134, 8.542076005864833592040376969577, 9.652981417897061763135296384406

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