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Properties

Label 2-209-19.11-c1-0-4
Degree $2$
Conductor $209$
Sign $-0.529 - 0.848i$
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  + (−0.198 + 0.343i)2-s + (−0.634 + 1.09i)3-s + (0.921 + 1.59i)4-s + (−0.220 + 0.382i)5-s + (−0.251 − 0.435i)6-s − 0.667·7-s − 1.52·8-s + (0.696 + 1.20i)9-s + (−0.0875 − 0.151i)10-s − 11-s − 2.33·12-s + (−0.802 − 1.38i)13-s + (0.132 − 0.229i)14-s + (−0.280 − 0.485i)15-s + (−1.54 + 2.66i)16-s + (−2.02 + 3.51i)17-s + ⋯
L(s)  = 1  + (−0.140 + 0.242i)2-s + (−0.366 + 0.634i)3-s + (0.460 + 0.797i)4-s + (−0.0988 + 0.171i)5-s + (−0.102 − 0.177i)6-s − 0.252·7-s − 0.538·8-s + (0.232 + 0.401i)9-s + (−0.0276 − 0.0479i)10-s − 0.301·11-s − 0.674·12-s + (−0.222 − 0.385i)13-s + (0.0353 − 0.0612i)14-s + (−0.0723 − 0.125i)15-s + (−0.385 + 0.667i)16-s + (−0.491 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.529 - 0.848i$
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: \((2,\ 209,\ (\ :1/2),\ -0.529 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477752 + 0.861308i\)
\(L(\frac12)\) \(\approx\) \(0.477752 + 0.861308i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.81 + 3.32i)T \)
good2 \( 1 + (0.198 - 0.343i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.634 - 1.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.220 - 0.382i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.667T + 7T^{2} \)
13 \( 1 + (0.802 + 1.38i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.02 - 3.51i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.187 - 0.324i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.28 - 2.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.01T + 31T^{2} \)
37 \( 1 - 7.24T + 37T^{2} \)
41 \( 1 + (-2.54 + 4.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.01 + 8.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.03 + 1.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.163 - 0.282i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.93 + 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.53 - 9.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.06 - 7.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.19 - 5.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.0659 + 0.114i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.134 - 0.233i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.76T + 83T^{2} \)
89 \( 1 + (0.492 + 0.853i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.68 + 2.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.72276655121431710033462652628, −11.56346293280217624556768830818, −10.84093633519639936730706022799, −9.882083471727343477847930693589, −8.659974857940747024945947922171, −7.63342420195934132399820151333, −6.70150726025516161285184985217, −5.35878531146718604610537979356, −4.04786716352010895999095385039, −2.68863653297730779824471600243, 0.961021465906418852558067755889, 2.65304923458753723921392827944, 4.64255758917027736631019846493, 6.05432106912669143953344634510, 6.71305987443738928247005272090, 7.88525147476114020666084240123, 9.409550956518242876091832490198, 10.02282096808592856093605306006, 11.30507024468356697933609623273, 11.91217749945376174185170216405

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