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Properties

Label 2-47e2-47.20-c0-0-0
Degree $2$
Conductor $2209$
Sign $-0.485 + 0.873i$
Analytic cond. $1.10243$
Root an. cond. $1.04996$
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  + (0.110 + 1.61i)2-s + (0.595 + 0.166i)3-s + (−1.60 + 0.220i)4-s + (−0.203 + 0.979i)6-s + (−1.10 + 1.18i)7-s + (−0.203 − 0.979i)8-s + (−0.528 − 0.321i)9-s + (−0.990 − 0.136i)12-s + (−2.03 − 1.65i)14-s + (−0.566 + 0.246i)17-s + (0.460 − 0.887i)18-s + (−0.854 + 0.519i)21-s + (0.0421 − 0.616i)24-s + (−0.334 + 0.942i)25-s + (−0.682 − 0.730i)27-s + (1.50 − 2.13i)28-s + ⋯
L(s)  = 1  + (0.110 + 1.61i)2-s + (0.595 + 0.166i)3-s + (−1.60 + 0.220i)4-s + (−0.203 + 0.979i)6-s + (−1.10 + 1.18i)7-s + (−0.203 − 0.979i)8-s + (−0.528 − 0.321i)9-s + (−0.990 − 0.136i)12-s + (−2.03 − 1.65i)14-s + (−0.566 + 0.246i)17-s + (0.460 − 0.887i)18-s + (−0.854 + 0.519i)21-s + (0.0421 − 0.616i)24-s + (−0.334 + 0.942i)25-s + (−0.682 − 0.730i)27-s + (1.50 − 2.13i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $-0.485 + 0.873i$
Analytic conductor: \(1.10243\)
Root analytic conductor: \(1.04996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2209,\ (\ :0),\ -0.485 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7852067893\)
\(L(\frac12)\) \(\approx\) \(0.7852067893\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.110 - 1.61i)T + (-0.990 + 0.136i)T^{2} \)
3 \( 1 + (-0.595 - 0.166i)T + (0.854 + 0.519i)T^{2} \)
5 \( 1 + (0.334 - 0.942i)T^{2} \)
7 \( 1 + (1.10 - 1.18i)T + (-0.0682 - 0.997i)T^{2} \)
11 \( 1 + (-0.682 - 0.730i)T^{2} \)
13 \( 1 + (0.576 - 0.816i)T^{2} \)
17 \( 1 + (0.566 - 0.246i)T + (0.682 - 0.730i)T^{2} \)
19 \( 1 + (0.334 + 0.942i)T^{2} \)
23 \( 1 + (0.990 + 0.136i)T^{2} \)
29 \( 1 + (0.576 + 0.816i)T^{2} \)
31 \( 1 + (-0.854 + 0.519i)T^{2} \)
37 \( 1 + (0.479 - 0.390i)T + (0.203 - 0.979i)T^{2} \)
41 \( 1 + (0.917 + 0.398i)T^{2} \)
43 \( 1 + (-0.962 + 0.269i)T^{2} \)
53 \( 1 + (0.329 - 1.58i)T + (-0.917 - 0.398i)T^{2} \)
59 \( 1 + (-1.60 - 0.220i)T + (0.962 + 0.269i)T^{2} \)
61 \( 1 + (-1.25 - 1.02i)T + (0.203 + 0.979i)T^{2} \)
67 \( 1 + (0.0682 - 0.997i)T^{2} \)
71 \( 1 + (0.0421 - 0.616i)T + (-0.990 - 0.136i)T^{2} \)
73 \( 1 + (-0.460 + 0.887i)T^{2} \)
79 \( 1 + (0.206 + 0.582i)T + (-0.775 + 0.631i)T^{2} \)
83 \( 1 + (1.83 + 0.796i)T + (0.682 + 0.730i)T^{2} \)
89 \( 1 + (-0.933 - 1.32i)T + (-0.334 + 0.942i)T^{2} \)
97 \( 1 + (1.55 + 0.436i)T + (0.854 + 0.519i)T^{2} \)
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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

−9.303170388828264855883258683921, −8.770698020440509071445949143532, −8.346291366492021581229036379938, −7.31197588963688317438873599436, −6.61408939655648569470102905472, −5.85617357398396736831780456008, −5.46465805051595029439007916377, −4.24473698169989236026614877259, −3.28577021133153188347393456544, −2.36502190186167351957337281014, 0.45560280880331593083016267684, 1.94854953328893833445528339582, 2.78430923381611499881646830908, 3.56423577164651864628946686690, 4.15306087956025743554633111253, 5.22000492070870134783867917125, 6.50398360361493473457053193235, 7.19718662381991973575723372927, 8.247935018513211550950569278023, 8.940048199611933380956110282755

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