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Properties

Label 2-2280-2280.1139-c0-0-9
Degree $2$
Conductor $2280$
Sign $1$
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  + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + 5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−0.707 − 0.707i)10-s − 2i·11-s + (−0.707 + 0.707i)12-s + 1.41i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (0.707 − 0.707i)18-s + 19-s + 1.00i·20-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + 5-s − 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−0.707 − 0.707i)10-s − 2i·11-s + (−0.707 + 0.707i)12-s + 1.41i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (0.707 − 0.707i)18-s + 19-s + 1.00i·20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.294380973\)
\(L(\frac12)\) \(\approx\) \(1.294380973\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 - T \)
19 \( 1 - T \)
good7 \( 1 + T^{2} \)
11 \( 1 + 2iT - T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−9.262456258777500743181124059876, −8.754787518646122551656230817877, −8.064484464735406014734094611762, −7.06951169140951198254663878238, −6.07321128985674711034621694424, −5.13438851989137499580098686019, −4.02912268570098613234240962597, −3.24258286672503804386024599498, −2.46662656194156765670634349321, −1.39257934592051679122159397248, 1.29009019022622477388278236017, 2.09731958829945289077175118154, 3.10005339381804278524581283226, 4.74549606019747834326616667564, 5.40828979053835392298701475311, 6.42780329122891501863328331842, 6.93861258916693822648808197311, 7.77200019795766434385753115491, 8.201721823764818544664508252990, 9.310452363529773999616143584486

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