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Properties

Label 2-209-19.11-c1-0-0
Degree 22
Conductor 209209
Sign 0.120+0.992i0.120 + 0.992i
Analytic cond. 1.668871.66887
Root an. cond. 1.291841.29184
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(209s/2 ΓC(s) L(s)=((0.120+0.992i) Λ‾(2−s)\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}
Λ(s)=(209s/2 ΓC(s+1/2) L(s)=((0.120+0.992i) Λ‾(1−s)\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: 22
Conductor: 209209    =    11⋅1911 \cdot 19
Sign: 0.120+0.992i0.120 + 0.992i
Analytic conductor: 1.668871.66887
Root analytic conductor: 1.291841.29184
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ209(144,⋅)\chi_{209} (144, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 209, ( :1/2), 0.120+0.992i)(2,\ 209,\ (\ :1/2),\ 0.120 + 0.992i)

Particular Values

L(1)L(1) ≈\approx 0.105934−0.0938910i0.105934 - 0.0938910i
L(12)L(\frac12) ≈\approx 0.105934−0.0938910i0.105934 - 0.0938910i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad11 1+T 1 + T
19 1+(3.96−1.82i)T 1 + (3.96 - 1.82i)T
good2 1+(0.918−1.59i)T+(−1−1.73i)T2 1 + (0.918 - 1.59i)T + (-1 - 1.73i)T^{2}
3 1+(1.56−2.70i)T+(−1.5−2.59i)T2 1 + (1.56 - 2.70i)T + (-1.5 - 2.59i)T^{2}
5 1+(−1.63+2.83i)T+(−2.5−4.33i)T2 1 + (-1.63 + 2.83i)T + (-2.5 - 4.33i)T^{2}
7 1+3.63T+7T2 1 + 3.63T + 7T^{2}
13 1+(−0.224−0.388i)T+(−6.5+11.2i)T2 1 + (-0.224 - 0.388i)T + (-6.5 + 11.2i)T^{2}
17 1+(2.11−3.66i)T+(−8.5−14.7i)T2 1 + (2.11 - 3.66i)T + (-8.5 - 14.7i)T^{2}
23 1+(−0.506−0.877i)T+(−11.5+19.9i)T2 1 + (-0.506 - 0.877i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.37−5.84i)T+(−14.5+25.1i)T2 1 + (-3.37 - 5.84i)T + (-14.5 + 25.1i)T^{2}
31 1+7.00T+31T2 1 + 7.00T + 31T^{2}
37 1+10.8T+37T2 1 + 10.8T + 37T^{2}
41 1+(−0.641+1.11i)T+(−20.5−35.5i)T2 1 + (-0.641 + 1.11i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2.08+3.60i)T+(−21.5−37.2i)T2 1 + (-2.08 + 3.60i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3.38−5.85i)T+(−23.5+40.7i)T2 1 + (-3.38 - 5.85i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.19+2.06i)T+(−26.5+45.8i)T2 1 + (1.19 + 2.06i)T + (-26.5 + 45.8i)T^{2}
59 1+(−4.75+8.23i)T+(−29.5−51.0i)T2 1 + (-4.75 + 8.23i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.33+4.03i)T+(−30.5+52.8i)T2 1 + (2.33 + 4.03i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.718−1.24i)T+(−33.5+58.0i)T2 1 + (-0.718 - 1.24i)T + (-33.5 + 58.0i)T^{2}
71 1+(5.28−9.14i)T+(−35.5−61.4i)T2 1 + (5.28 - 9.14i)T + (-35.5 - 61.4i)T^{2}
73 1+(1.32−2.29i)T+(−36.5−63.2i)T2 1 + (1.32 - 2.29i)T + (-36.5 - 63.2i)T^{2}
79 1+(−0.230+0.399i)T+(−39.5−68.4i)T2 1 + (-0.230 + 0.399i)T + (-39.5 - 68.4i)T^{2}
83 1+3.09T+83T2 1 + 3.09T + 83T^{2}
89 1+(−2.31−4.00i)T+(−44.5+77.0i)T2 1 + (-2.31 - 4.00i)T + (-44.5 + 77.0i)T^{2}
97 1+(4.62−8.01i)T+(−48.5−84.0i)T2 1 + (4.62 - 8.01i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line