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Properties

Label 2-209-11.5-c1-0-13
Degree $2$
Conductor $209$
Sign $0.733 + 0.679i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.262 − 0.808i)2-s + (1.60 + 1.16i)3-s + (1.03 − 0.751i)4-s + (0.485 − 1.49i)5-s + (0.520 − 1.60i)6-s + (0.244 − 0.177i)7-s + (−2.25 − 1.63i)8-s + (0.286 + 0.881i)9-s − 1.33·10-s + (−2.13 + 2.53i)11-s + 2.53·12-s + (0.200 + 0.617i)13-s + (−0.208 − 0.151i)14-s + (2.51 − 1.83i)15-s + (0.0582 − 0.179i)16-s + (−1.46 + 4.52i)17-s + ⋯
L(s)  = 1  + (−0.185 − 0.571i)2-s + (0.925 + 0.672i)3-s + (0.516 − 0.375i)4-s + (0.217 − 0.668i)5-s + (0.212 − 0.653i)6-s + (0.0925 − 0.0672i)7-s + (−0.796 − 0.578i)8-s + (0.0954 + 0.293i)9-s − 0.422·10-s + (−0.645 + 0.764i)11-s + 0.731·12-s + (0.0556 + 0.171i)13-s + (−0.0556 − 0.0404i)14-s + (0.650 − 0.472i)15-s + (0.0145 − 0.0448i)16-s + (−0.356 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ 0.733 + 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47803 - 0.579244i\)
\(L(\frac12)\) \(\approx\) \(1.47803 - 0.579244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.13 - 2.53i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (0.262 + 0.808i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.60 - 1.16i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.485 + 1.49i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.244 + 0.177i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.200 - 0.617i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.46 - 4.52i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 - 2.02T + 23T^{2} \)
29 \( 1 + (4.06 - 2.95i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.866 + 2.66i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.64 - 1.92i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.20 + 1.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.53T + 43T^{2} \)
47 \( 1 + (-3.09 - 2.24i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.687 + 2.11i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.48 - 6.16i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.59 + 4.90i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.16T + 67T^{2} \)
71 \( 1 + (-2.20 + 6.77i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.19 - 5.22i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.15 + 6.63i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.10 + 15.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (0.0370 + 0.114i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.32702147438678381734111431748, −11.04238137596277540734292341310, −10.23463394452735312710679665655, −9.383138283302304640873121431319, −8.694055163837230901748378404318, −7.36283958759230571047223506803, −5.90405981764799685113820562862, −4.52502244719431006299798590104, −3.17044653630825752363183866986, −1.80134163953148810632052381498, 2.40557699609627850505694123256, 3.15951708964243476656995906626, 5.45002319376026026414302143889, 6.73477110368851389031965504231, 7.44345336476490060282459287534, 8.279450689751687457446153856256, 9.134161363245203920250355851403, 10.67695472206620470069599804234, 11.47100482120961325576249086025, 12.68987733926370457738210674840

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