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Properties

Label 2-2289-2289.1640-c0-0-0
Degree $2$
Conductor $2289$
Sign $0.303 + 0.952i$
Analytic cond. $1.14235$
Root an. cond. $1.06881$
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  + (0.973 − 0.230i)3-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)7-s + (0.893 − 0.448i)9-s + (−0.0581 − 0.998i)12-s + (0.310 + 0.155i)13-s + (−0.939 − 0.342i)16-s + (0.337 + 1.91i)19-s + (−0.0581 − 0.998i)21-s + (0.597 + 0.802i)25-s + (0.766 − 0.642i)27-s + (−0.939 − 0.342i)28-s + (−0.290 − 0.190i)31-s + (−0.286 − 0.957i)36-s + (−1.49 − 0.982i)37-s + ⋯
L(s)  = 1  + (0.973 − 0.230i)3-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)7-s + (0.893 − 0.448i)9-s + (−0.0581 − 0.998i)12-s + (0.310 + 0.155i)13-s + (−0.939 − 0.342i)16-s + (0.337 + 1.91i)19-s + (−0.0581 − 0.998i)21-s + (0.597 + 0.802i)25-s + (0.766 − 0.642i)27-s + (−0.939 − 0.342i)28-s + (−0.290 − 0.190i)31-s + (−0.286 − 0.957i)36-s + (−1.49 − 0.982i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2289\)    =    \(3 \cdot 7 \cdot 109\)
Sign: $0.303 + 0.952i$
Analytic conductor: \(1.14235\)
Root analytic conductor: \(1.06881\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2289} (1640, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2289,\ (\ :0),\ 0.303 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.791997372\)
\(L(\frac12)\) \(\approx\) \(1.791997372\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.973 + 0.230i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
109 \( 1 + (-0.766 - 0.642i)T \)
good2 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.597 - 0.802i)T^{2} \)
11 \( 1 + (0.0581 + 0.998i)T^{2} \)
13 \( 1 + (-0.310 - 0.155i)T + (0.597 + 0.802i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.337 - 1.91i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.893 + 0.448i)T^{2} \)
31 \( 1 + (0.290 + 0.190i)T + (0.396 + 0.918i)T^{2} \)
37 \( 1 + (1.49 + 0.982i)T + (0.396 + 0.918i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.12 + 0.408i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.686 + 0.727i)T^{2} \)
53 \( 1 + (-0.396 + 0.918i)T^{2} \)
59 \( 1 + (0.835 - 0.549i)T^{2} \)
61 \( 1 + (0.997 - 1.34i)T + (-0.286 - 0.957i)T^{2} \)
67 \( 1 + (-0.835 - 0.549i)T + (0.396 + 0.918i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.36 + 1.44i)T + (-0.0581 - 0.998i)T^{2} \)
79 \( 1 + (1.77 - 0.891i)T + (0.597 - 0.802i)T^{2} \)
83 \( 1 + (0.0581 - 0.998i)T^{2} \)
89 \( 1 + (0.686 - 0.727i)T^{2} \)
97 \( 1 + (0.290 - 0.190i)T + (0.396 - 0.918i)T^{2} \)
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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

−9.056836423061018267020068686815, −8.296042018984142169108632561167, −7.41851340870054083866022991542, −6.92046860440809121221195899223, −5.98079861224280824397683005454, −5.07316370085531211281083989466, −4.00897187724016274465882838925, −3.35681544152159951160167876119, −1.93034776746859383060454609709, −1.25962490728374376015496605027, 1.90393000119383488221084067965, 2.85354652324858345641394249224, 3.30565346727037905336551933026, 4.51954190119187198342210493660, 5.12553791701655276847876084082, 6.55419540713772451278466245322, 7.09920929209643825197709991894, 8.098739777682071781603463010234, 8.563488281826966495039104331425, 9.069375968068963259051214555970

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