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Properties

Label 2-2280-2280.149-c0-0-2
Degree $2$
Conductor $2280$
Sign $-0.356 - 0.934i$
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 + (0.342 + 0.939i)3-s + (0.499 + 0.866i)4-s + (0.984 + 0.173i)5-s + (−0.173 + 0.984i)6-s + 0.999i·8-s + (−0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−0.642 + 0.766i)12-s + (0.173 + 0.984i)15-s + (−0.5 + 0.866i)16-s + (−0.984 − 0.826i)17-s + (−0.984 + 0.173i)18-s + (0.939 − 0.342i)19-s + (0.342 + 0.939i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.342 + 0.939i)3-s + (0.499 + 0.866i)4-s + (0.984 + 0.173i)5-s + (−0.173 + 0.984i)6-s + 0.999i·8-s + (−0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−0.642 + 0.766i)12-s + (0.173 + 0.984i)15-s + (−0.5 + 0.866i)16-s + (−0.984 − 0.826i)17-s + (−0.984 + 0.173i)18-s + (0.939 − 0.342i)19-s + (0.342 + 0.939i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.356 - 0.934i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ -0.356 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.461598925\)
\(L(\frac12)\) \(\approx\) \(2.461598925\)
\(L(1)\) 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 \( 1 + (-0.342 - 0.939i)T \)
5 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.984 + 0.826i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.300 + 1.70i)T + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (1.50 - 1.26i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (-1.50 + 0.266i)T + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−9.399313899642174710785059268439, −8.728064614999930242179623753277, −7.87490611006388416721084217075, −6.91651388262979412051576383986, −6.18609382587339390639499307910, −5.36045570161972751336172682130, −4.74112316356365559276468352395, −3.92988207697281152872630394166, −2.79730037765058146652185456382, −2.28795269865614845689454342154, 1.44231220797025604146390422432, 1.96651756095480005640646469026, 3.07952420133968703780168928538, 3.86689627186954965792043004105, 5.33187860016042714854797281223, 5.57684232631362716036573898427, 6.67008742371356134207488058035, 7.03984949762457948554558267362, 8.228632700090879533218347987776, 9.057133642723107935778576871246

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