Properties

Label 2-11-11.8-c2-0-0
Degree $2$
Conductor $11$
Sign $0.751 + 0.659i$
Analytic cond. $0.299728$
Root an. cond. $0.547474$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 − 2.48i)2-s + (1.11 + 3.44i)3-s + (−1.69 + 5.20i)4-s + (−3.23 − 2.35i)5-s + (6.54 − 9.00i)6-s + (−0.854 − 0.277i)7-s + (4.30 − 1.40i)8-s + (−3.30 + 2.40i)9-s + 12.3i·10-s + (10.8 − 1.67i)11-s − 19.7·12-s + (−5 − 6.88i)13-s + (0.854 + 2.62i)14-s + (4.47 − 13.7i)15-s + (6.42 + 4.66i)16-s + (−14.5 + 20.0i)17-s + ⋯
L(s)  = 1  + (−0.904 − 1.24i)2-s + (0.372 + 1.14i)3-s + (−0.422 + 1.30i)4-s + (−0.647 − 0.470i)5-s + (1.09 − 1.50i)6-s + (−0.122 − 0.0396i)7-s + (0.538 − 0.175i)8-s + (−0.367 + 0.267i)9-s + 1.23i·10-s + (0.988 − 0.152i)11-s − 1.64·12-s + (−0.384 − 0.529i)13-s + (0.0610 + 0.187i)14-s + (0.298 − 0.917i)15-s + (0.401 + 0.291i)16-s + (−0.854 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(0.299728\)
Root analytic conductor: \(0.547474\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :1),\ 0.751 + 0.659i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.467593 - 0.175960i\)
\(L(\frac12)\) \(\approx\) \(0.467593 - 0.175960i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-10.8 + 1.67i)T \)
good2 \( 1 + (1.80 + 2.48i)T + (-1.23 + 3.80i)T^{2} \)
3 \( 1 + (-1.11 - 3.44i)T + (-7.28 + 5.29i)T^{2} \)
5 \( 1 + (3.23 + 2.35i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (0.854 + 0.277i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (5 + 6.88i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (14.5 - 20.0i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-11.2 + 3.66i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 7.23T + 529T^{2} \)
29 \( 1 + (3.29 + 1.06i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (26.7 - 19.4i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-12.4 + 38.2i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-1.24 + 0.403i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 33.0iT - 1.84e3T^{2} \)
47 \( 1 + (-7.03 - 21.6i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-63.5 + 46.1i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (-9.66 + 29.7i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (16.5 - 22.7i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 76.5T + 4.48e3T^{2} \)
71 \( 1 + (-50.4 - 36.6i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-89.5 - 29.0i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-38.7 - 53.3i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-33.3 + 45.8i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 62.2T + 7.92e3T^{2} \)
97 \( 1 + (58.5 - 42.5i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−19.94904288129747247488059750311, −19.69972901695054936116869288575, −17.79222752320516828570737513128, −16.32613063810895925261879876950, −14.91052484225163681840645908379, −12.47605149836903241195723430943, −11.02728797715383661094363382787, −9.688702513640343836898559977297, −8.557336269125950525672707067218, −3.85750981494506242316922035002, 6.75128768761676733105855494842, 7.60075896120508420027768281865, 9.239798527400808300795729658819, 11.87472738208697920816204857607, 13.89344913458055276651457442668, 15.20094616670566885973205042721, 16.61418336368407249856808449338, 18.05015108541803500062783647920, 18.86450003332732086217879824181, 19.88451506168042867228387760183

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