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Properties

Label 2-209-19.11-c1-0-0
Degree $2$
Conductor $209$
Sign $0.120 + 0.992i$
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  + (−0.918 + 1.59i)2-s + (−1.56 + 2.70i)3-s + (−0.688 − 1.19i)4-s + (1.63 − 2.83i)5-s + (−2.87 − 4.97i)6-s − 3.63·7-s − 1.14·8-s + (−3.38 − 5.87i)9-s + (3.00 + 5.20i)10-s − 11-s + 4.30·12-s + (0.224 + 0.388i)13-s + (3.33 − 5.77i)14-s + (5.11 + 8.85i)15-s + (2.42 − 4.20i)16-s + (−2.11 + 3.66i)17-s + ⋯
L(s)  = 1  + (−0.649 + 1.12i)2-s + (−0.902 + 1.56i)3-s + (−0.344 − 0.595i)4-s + (0.731 − 1.26i)5-s + (−1.17 − 2.03i)6-s − 1.37·7-s − 0.405·8-s + (−1.12 − 1.95i)9-s + (0.949 + 1.64i)10-s − 0.301·11-s + 1.24·12-s + (0.0621 + 0.107i)13-s + (0.891 − 1.54i)14-s + (1.32 + 2.28i)15-s + (0.607 − 1.05i)16-s + (−0.512 + 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.120 + 0.992i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ 0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105934 - 0.0938910i\)
\(L(\frac12)\) \(\approx\) \(0.105934 - 0.0938910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
19 \( 1 + (3.96 - 1.82i)T \)
good2 \( 1 + (0.918 - 1.59i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.56 - 2.70i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.63 + 2.83i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.63T + 7T^{2} \)
13 \( 1 + (-0.224 - 0.388i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.11 - 3.66i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.506 - 0.877i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.37 - 5.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.00T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + (-0.641 + 1.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.08 + 3.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.38 - 5.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.19 + 2.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.75 + 8.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.33 + 4.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.718 - 1.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.28 - 9.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.32 - 2.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.230 + 0.399i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.09T + 83T^{2} \)
89 \( 1 + (-2.31 - 4.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.62 - 8.01i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.88425339988613467635774702845, −12.30823223822713869546808292250, −10.70152489036232728213168826346, −9.928781232225153702679992347634, −9.105687346626292575085285555703, −8.661989602240053045730393880133, −6.67737398214200015986501890388, −5.86381448876870529883472179922, −5.12604767183538157008783791667, −3.70402230011527282079545244075, 0.15212101577789393005132223629, 2.13276446198460661754991724304, 2.93426154206872820183079901465, 5.83621902594906652700621969632, 6.52856633379932823753212627282, 7.22769160759113462316127147029, 8.896428051201244487045764871720, 10.11201559061454038498668570335, 10.71510828655367504241868430420, 11.53781679538705677767299210769

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