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Properties

Label 2-209-11.3-c1-0-10
Degree $2$
Conductor $209$
Sign $0.328 + 0.944i$
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  + (−0.732 + 0.531i)2-s + (−0.173 − 0.533i)3-s + (−0.364 + 1.12i)4-s + (−2.22 − 1.61i)5-s + (0.410 + 0.298i)6-s + (0.170 − 0.524i)7-s + (−0.889 − 2.73i)8-s + (2.17 − 1.57i)9-s + 2.49·10-s + (0.885 − 3.19i)11-s + 0.662·12-s + (1.24 − 0.907i)13-s + (0.154 + 0.474i)14-s + (−0.477 + 1.46i)15-s + (0.197 + 0.143i)16-s + (−1.58 − 1.15i)17-s + ⋯
L(s)  = 1  + (−0.517 + 0.376i)2-s + (−0.100 − 0.308i)3-s + (−0.182 + 0.561i)4-s + (−0.996 − 0.723i)5-s + (0.167 + 0.121i)6-s + (0.0643 − 0.198i)7-s + (−0.314 − 0.968i)8-s + (0.724 − 0.526i)9-s + 0.787·10-s + (0.266 − 0.963i)11-s + 0.191·12-s + (0.346 − 0.251i)13-s + (0.0411 + 0.126i)14-s + (−0.123 + 0.379i)15-s + (0.0492 + 0.0357i)16-s + (−0.385 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :1/2),\ 0.328 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506653 - 0.360363i\)
\(L(\frac12)\) \(\approx\) \(0.506653 - 0.360363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.885 + 3.19i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.732 - 0.531i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.173 + 0.533i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.22 + 1.61i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.170 + 0.524i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.24 + 0.907i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.58 + 1.15i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + 3.43T + 23T^{2} \)
29 \( 1 + (-0.396 + 1.22i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.51 + 2.55i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.30 - 7.08i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.17 + 6.69i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.87T + 43T^{2} \)
47 \( 1 + (-1.91 - 5.90i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.22 + 2.34i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.728 - 2.24i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.84 + 1.33i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 + (-9.70 - 7.04i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.0934 - 0.287i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.46 - 1.06i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.64 + 6.28i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (-15.5 + 11.3i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.13986524084791433981733301602, −11.52009724773554520467823800537, −10.03537913099160393269683484897, −8.846900897890666170969960556344, −8.236358582465348303414935361255, −7.32966121266913446123770478977, −6.28501779821003610170311425321, −4.44797293641431839768089365219, −3.53893331239534501116496076531, −0.66801198746265703904286002414, 1.96669087927936170782408820516, 3.88798091614595740286170369969, 4.98980988779193609713319442567, 6.54958257414903233849705274708, 7.63805798275528920844434959046, 8.736022889620701719520806322633, 9.911328480329536488162143673033, 10.52805864850889601407068236968, 11.38818708054631227661628844368, 12.21472189128733799141227173533

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