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Properties

Label 2-2289-2289.443-c0-0-0
Degree $2$
Conductor $2289$
Sign $0.289 + 0.957i$
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  + (−0.286 + 0.957i)3-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)7-s + (−0.835 − 0.549i)9-s + (0.893 + 0.448i)12-s + (−0.290 + 0.190i)13-s + (−0.939 − 0.342i)16-s + (−0.0996 − 0.564i)19-s + (0.893 + 0.448i)21-s + (0.396 − 0.918i)25-s + (0.766 − 0.642i)27-s + (−0.939 − 0.342i)28-s + (−0.0201 + 0.346i)31-s + (−0.686 + 0.727i)36-s + (0.0971 − 1.66i)37-s + ⋯
L(s)  = 1  + (−0.286 + 0.957i)3-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)7-s + (−0.835 − 0.549i)9-s + (0.893 + 0.448i)12-s + (−0.290 + 0.190i)13-s + (−0.939 − 0.342i)16-s + (−0.0996 − 0.564i)19-s + (0.893 + 0.448i)21-s + (0.396 − 0.918i)25-s + (0.766 − 0.642i)27-s + (−0.939 − 0.342i)28-s + (−0.0201 + 0.346i)31-s + (−0.686 + 0.727i)36-s + (0.0971 − 1.66i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2289\)    =    \(3 \cdot 7 \cdot 109\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(1.14235\)
Root analytic conductor: \(1.06881\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2289} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2289,\ (\ :0),\ 0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9273640492\)
\(L(\frac12)\) \(\approx\) \(0.9273640492\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.286 - 0.957i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
109 \( 1 + (-0.766 - 0.642i)T \)
good2 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.396 + 0.918i)T^{2} \)
11 \( 1 + (-0.893 - 0.448i)T^{2} \)
13 \( 1 + (0.290 - 0.190i)T + (0.396 - 0.918i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 + (0.0996 + 0.564i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.835 + 0.549i)T^{2} \)
31 \( 1 + (0.0201 - 0.346i)T + (-0.993 - 0.116i)T^{2} \)
37 \( 1 + (-0.0971 + 1.66i)T + (-0.993 - 0.116i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.744 + 0.270i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.973 + 0.230i)T^{2} \)
53 \( 1 + (0.993 - 0.116i)T^{2} \)
59 \( 1 + (0.0581 + 0.998i)T^{2} \)
61 \( 1 + (0.0460 + 0.106i)T + (-0.686 + 0.727i)T^{2} \)
67 \( 1 + (-0.0581 + 0.998i)T + (-0.993 - 0.116i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.16 - 0.275i)T + (0.893 + 0.448i)T^{2} \)
79 \( 1 + (0.997 + 0.656i)T + (0.396 + 0.918i)T^{2} \)
83 \( 1 + (-0.893 + 0.448i)T^{2} \)
89 \( 1 + (-0.973 - 0.230i)T^{2} \)
97 \( 1 + (0.0201 + 0.346i)T + (-0.993 + 0.116i)T^{2} \)
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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

−9.258378718499630964494091660338, −8.484934826374832386133851629517, −7.33401034538034886283718111230, −6.61298041687723867326816478504, −5.83619792876034355674725879756, −4.90945730289874132085132833403, −4.45110377905618545908025923278, −3.43103387797352809613652275104, −2.16426489823793055244414835818, −0.64964043184115249646884497766, 1.64427755067883202125168330540, 2.60808902523626035646234173540, 3.34295714362534096057882178716, 4.69122175999770770753857879900, 5.53513556080720908469967198862, 6.37243512127861958921765361272, 7.07757637121931115039244304921, 7.88777565613972665527280700293, 8.372063358334403609515899138033, 9.059659347830563528496891276763

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