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Properties

Label 2-209-19.11-c1-0-7
Degree $2$
Conductor $209$
Sign $0.164 - 0.986i$
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  + (−1.34 + 2.33i)2-s + (0.0468 − 0.0812i)3-s + (−2.63 − 4.56i)4-s + (1.12 − 1.94i)5-s + (0.126 + 0.218i)6-s + 0.215·7-s + 8.80·8-s + (1.49 + 2.59i)9-s + (3.02 + 5.24i)10-s − 11-s − 0.493·12-s + (2.79 + 4.84i)13-s + (−0.290 + 0.503i)14-s + (−0.105 − 0.182i)15-s + (−6.60 + 11.4i)16-s + (3.92 − 6.80i)17-s + ⋯
L(s)  = 1  + (−0.953 + 1.65i)2-s + (0.0270 − 0.0468i)3-s + (−1.31 − 2.28i)4-s + (0.501 − 0.869i)5-s + (0.0515 + 0.0893i)6-s + 0.0815·7-s + 3.11·8-s + (0.498 + 0.863i)9-s + (0.956 + 1.65i)10-s − 0.301·11-s − 0.142·12-s + (0.775 + 1.34i)13-s + (−0.0777 + 0.134i)14-s + (−0.0271 − 0.0470i)15-s + (−1.65 + 2.85i)16-s + (0.952 − 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.164 - 0.986i$
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: \((2,\ 209,\ (\ :1/2),\ 0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626553 + 0.530641i\)
\(L(\frac12)\) \(\approx\) \(0.626553 + 0.530641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.31 + 0.631i)T \)
good2 \( 1 + (1.34 - 2.33i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0468 + 0.0812i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.12 + 1.94i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.215T + 7T^{2} \)
13 \( 1 + (-2.79 - 4.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.92 + 6.80i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.21 - 3.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 - 1.18T + 37T^{2} \)
41 \( 1 + (2.15 - 3.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.239 + 0.414i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.115 - 0.199i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.78 + 8.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.86 + 8.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.73 - 6.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.29 + 7.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.03 - 5.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.53 - 4.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.16 + 2.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + (-0.252 - 0.437i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.84 + 3.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.04665061871349912257543656613, −11.40105135251601055966487025328, −10.03548213417954709590288268595, −9.389848031016604080101187065028, −8.537995884688652476151193341009, −7.57273230654705059553078575077, −6.69808125156802037982299531993, −5.36496605014477519141867723774, −4.77628480869362314043864252753, −1.36393191557054089880317300792, 1.35536163171418675715166024279, 3.01284847871458178743739620617, 3.79031298682607870707793649226, 5.90800576661001499232510158277, 7.57671706352239859527200383552, 8.430952614130418807562779417352, 9.718365793003935373189596393569, 10.23526654448019599081690517939, 10.88951690927263277060764083431, 12.01302503573835430222238371342

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