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Properties

Label 2-209-11.2-c2-0-8
Degree $2$
Conductor $209$
Sign $0.794 - 0.607i$
Analytic cond. $5.69483$
Root an. cond. $2.38638$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 − 0.553i)2-s + (1.61 − 1.17i)3-s + (−0.644 − 0.468i)4-s + (−0.908 − 2.79i)5-s + (−3.39 + 1.10i)6-s + (−6.55 + 9.02i)7-s + (5.04 + 6.94i)8-s + (−1.54 + 4.77i)9-s + 5.26i·10-s + (9.19 + 6.04i)11-s − 1.59·12-s + (0.181 + 0.0588i)13-s + (16.1 − 11.7i)14-s + (−4.75 − 3.45i)15-s + (−3.76 − 11.5i)16-s + (24.9 − 8.10i)17-s + ⋯
L(s)  = 1  + (−0.851 − 0.276i)2-s + (0.538 − 0.391i)3-s + (−0.161 − 0.117i)4-s + (−0.181 − 0.559i)5-s + (−0.566 + 0.183i)6-s + (−0.936 + 1.28i)7-s + (0.630 + 0.868i)8-s + (−0.172 + 0.530i)9-s + 0.526i·10-s + (0.835 + 0.549i)11-s − 0.132·12-s + (0.0139 + 0.00452i)13-s + (1.15 − 0.838i)14-s + (−0.316 − 0.230i)15-s + (−0.235 − 0.723i)16-s + (1.46 − 0.476i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(5.69483\)
Root analytic conductor: \(2.38638\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1),\ 0.794 - 0.607i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.773614 + 0.262012i\)
\(L(\frac12)\) \(\approx\) \(0.773614 + 0.262012i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-9.19 - 6.04i)T \)
19 \( 1 + (-2.56 - 3.52i)T \)
good2 \( 1 + (1.70 + 0.553i)T + (3.23 + 2.35i)T^{2} \)
3 \( 1 + (-1.61 + 1.17i)T + (2.78 - 8.55i)T^{2} \)
5 \( 1 + (0.908 + 2.79i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (6.55 - 9.02i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (-0.181 - 0.0588i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-24.9 + 8.10i)T + (233. - 169. i)T^{2} \)
23 \( 1 + 20.5T + 529T^{2} \)
29 \( 1 + (28.6 - 39.4i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-3.94 + 12.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-31.9 - 23.2i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-25.7 - 35.5i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 54.9iT - 1.84e3T^{2} \)
47 \( 1 + (58.2 - 42.3i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (16.8 - 51.9i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (61.6 + 44.7i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-48.7 + 15.8i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 7.56T + 4.48e3T^{2} \)
71 \( 1 + (-21.3 - 65.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-1.09 + 1.50i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (70.8 + 23.0i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-149. + 48.5i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 3.78T + 7.92e3T^{2} \)
97 \( 1 + (-17.1 + 52.7i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−12.31343663901920408717512879317, −11.28274456721142295950298843191, −9.775407322772142444476627908502, −9.382231678978903205129715834122, −8.436752848245314136894997252827, −7.65464804412377732609589699059, −6.03471499360140160310315518035, −4.86435162995442908155822334166, −2.93629290947201719308871428107, −1.49696309443022929283947676602, 0.63032858351204350457453186319, 3.56758970444284815078278415413, 3.82106688022321256995532268426, 6.21289341526599913253583773506, 7.19167564456497596552832460690, 8.073101696032584374753627622165, 9.237347366027618359568611415813, 9.853925409409370740287146790066, 10.62140135330280017515043937619, 11.98813274689993740523765282247

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