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Properties

Label 2-209-11.3-c1-0-4
Degree $2$
Conductor $209$
Sign $0.896 - 0.443i$
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  + (−1.18 + 0.864i)2-s + (−0.639 − 1.96i)3-s + (0.0502 − 0.154i)4-s + (0.989 + 0.718i)5-s + (2.46 + 1.78i)6-s + (−0.728 + 2.24i)7-s + (−0.834 − 2.56i)8-s + (−1.03 + 0.753i)9-s − 1.79·10-s + (3.28 + 0.479i)11-s − 0.336·12-s + (2.06 − 1.50i)13-s + (−1.07 − 3.29i)14-s + (0.782 − 2.40i)15-s + (3.47 + 2.52i)16-s + (5.81 + 4.22i)17-s + ⋯
L(s)  = 1  + (−0.841 + 0.611i)2-s + (−0.369 − 1.13i)3-s + (0.0251 − 0.0773i)4-s + (0.442 + 0.321i)5-s + (1.00 + 0.730i)6-s + (−0.275 + 0.847i)7-s + (−0.295 − 0.908i)8-s + (−0.345 + 0.251i)9-s − 0.568·10-s + (0.989 + 0.144i)11-s − 0.0971·12-s + (0.573 − 0.416i)13-s + (−0.286 − 0.881i)14-s + (0.201 − 0.621i)15-s + (0.869 + 0.631i)16-s + (1.40 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.896 - 0.443i$
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: \((2,\ 209,\ (\ :1/2),\ 0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749569 + 0.175321i\)
\(L(\frac12)\) \(\approx\) \(0.749569 + 0.175321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.28 - 0.479i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (1.18 - 0.864i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.639 + 1.96i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.989 - 0.718i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.728 - 2.24i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.06 + 1.50i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.81 - 4.22i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 - 6.34T + 23T^{2} \)
29 \( 1 + (1.34 - 4.13i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.49 - 2.53i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.25 + 6.93i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.36 - 10.3i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.28T + 43T^{2} \)
47 \( 1 + (3.66 + 11.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.504 - 0.366i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.0443 + 0.136i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.33 + 3.15i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.80T + 67T^{2} \)
71 \( 1 + (2.65 + 1.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.682 - 2.09i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.95 - 5.05i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.83 + 3.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + (-1.12 + 0.814i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.59471000201501407332134697401, −11.69131270631688254814310135142, −10.26536913102522906447413955310, −9.257108602961692844448645886600, −8.361562022966260147223592632114, −7.34291258980162346704368768257, −6.44261365935631967519894617458, −5.80363309874037569977914708919, −3.41968186120199046747053025971, −1.38694841499191491090714084908, 1.19486150149946947123931596337, 3.44928968536212069288417111523, 4.77867910442720727523813922218, 5.84663296464850856499260815033, 7.40577270716483326369156306120, 9.017275837066384870314198854774, 9.510060487535669508431034425510, 10.19872300734712040257834668017, 11.11886071658120121042360608876, 11.73245205186603294801165260099

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