Properties

Label 2-11-11.3-c5-0-1
Degree $2$
Conductor $11$
Sign $-0.613 + 0.789i$
Analytic cond. $1.76422$
Root an. cond. $1.32824$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.36 + 4.62i)2-s + (−5.98 − 18.4i)3-s + (9.23 − 28.4i)4-s + (−59.3 − 43.1i)5-s + (123. + 89.5i)6-s + (−51.6 + 158. i)7-s + (−5.17 − 15.9i)8-s + (−106. + 77.6i)9-s + 576.·10-s + (3.42 − 401. i)11-s − 578.·12-s + (−139. + 101. i)13-s + (−405. − 1.24e3i)14-s + (−438. + 1.35e3i)15-s + (879. + 639. i)16-s + (365. + 265. i)17-s + ⋯
L(s)  = 1  + (−1.12 + 0.817i)2-s + (−0.383 − 1.18i)3-s + (0.288 − 0.887i)4-s + (−1.06 − 0.771i)5-s + (1.39 + 1.01i)6-s + (−0.398 + 1.22i)7-s + (−0.0285 − 0.0879i)8-s + (−0.439 + 0.319i)9-s + 1.82·10-s + (0.00853 − 0.999i)11-s − 1.15·12-s + (−0.228 + 0.166i)13-s + (−0.553 − 1.70i)14-s + (−0.503 + 1.55i)15-s + (0.859 + 0.624i)16-s + (0.306 + 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.613 + 0.789i$
Analytic conductor: \(1.76422\)
Root analytic conductor: \(1.32824\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :5/2),\ -0.613 + 0.789i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.108914 - 0.222573i\)
\(L(\frac12)\) \(\approx\) \(0.108914 - 0.222573i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.42 + 401. i)T \)
good2 \( 1 + (6.36 - 4.62i)T + (9.88 - 30.4i)T^{2} \)
3 \( 1 + (5.98 + 18.4i)T + (-196. + 142. i)T^{2} \)
5 \( 1 + (59.3 + 43.1i)T + (965. + 2.97e3i)T^{2} \)
7 \( 1 + (51.6 - 158. i)T + (-1.35e4 - 9.87e3i)T^{2} \)
13 \( 1 + (139. - 101. i)T + (1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (-365. - 265. i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (764. + 2.35e3i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 + 2.58e3T + 6.43e6T^{2} \)
29 \( 1 + (709. - 2.18e3i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (-3.78e3 + 2.74e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (-951. + 2.92e3i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (-2.21e3 - 6.81e3i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 - 398.T + 1.47e8T^{2} \)
47 \( 1 + (-415. - 1.27e3i)T + (-1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (8.16e3 - 5.93e3i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (-1.22e4 + 3.77e4i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (-1.36e4 - 9.89e3i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 + 2.89e4T + 1.35e9T^{2} \)
71 \( 1 + (4.91e4 + 3.57e4i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (-1.10e4 + 3.39e4i)T + (-1.67e9 - 1.21e9i)T^{2} \)
79 \( 1 + (-1.01e4 + 7.35e3i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (-1.22e4 - 8.91e3i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 - 1.01e5T + 5.58e9T^{2} \)
97 \( 1 + (-1.08e5 + 7.87e4i)T + (2.65e9 - 8.16e9i)T^{2} \)
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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

−18.91302809011375762882592332773, −17.77031815464258247633934634369, −16.39850517838538335826718582864, −15.46063543348171096048024396910, −12.85732854987115256156920710611, −11.78192233575174191278302716776, −8.963976807570103129485698927172, −7.87714962784561615865364335275, −6.24879423684032138894150288349, −0.35262732083598023687836627605, 3.91137272689274126697257756700, 7.64592495914858090483041005725, 10.01100456787731917301877102256, 10.47574952505201901448061094339, 11.88887381011350717624838039602, 14.71094343531759850434564794099, 16.15765046817417820432463062719, 17.32945423960902489542752558722, 18.88133451372181988914635868445, 19.97160449697029364981980689209

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