Properties

Label 2-11-11.2-c12-0-3
Degree $2$
Conductor $11$
Sign $0.996 + 0.0877i$
Analytic cond. $10.0539$
Root an. cond. $3.17079$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−92.8 − 30.1i)2-s + (691. − 502. i)3-s + (4.39e3 + 3.19e3i)4-s + (1.08e3 + 3.33e3i)5-s + (−7.93e4 + 2.57e4i)6-s + (−6.92e4 + 9.52e4i)7-s + (−7.64e4 − 1.05e5i)8-s + (6.14e4 − 1.89e5i)9-s − 3.42e5i·10-s + (1.64e6 + 6.57e5i)11-s + 4.63e6·12-s + (4.52e6 + 1.47e6i)13-s + (9.29e6 − 6.75e6i)14-s + (2.42e6 + 1.76e6i)15-s + (−2.94e6 − 9.07e6i)16-s + (−7.78e6 + 2.52e6i)17-s + ⋯
L(s)  = 1  + (−1.45 − 0.471i)2-s + (0.948 − 0.688i)3-s + (1.07 + 0.779i)4-s + (0.0693 + 0.213i)5-s + (−1.69 + 0.552i)6-s + (−0.588 + 0.809i)7-s + (−0.291 − 0.401i)8-s + (0.115 − 0.355i)9-s − 0.342i·10-s + (0.928 + 0.370i)11-s + 1.55·12-s + (0.937 + 0.304i)13-s + (1.23 − 0.897i)14-s + (0.212 + 0.154i)15-s + (−0.175 − 0.540i)16-s + (−0.322 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0877i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.996 + 0.0877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.996 + 0.0877i$
Analytic conductor: \(10.0539\)
Root analytic conductor: \(3.17079\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :6),\ 0.996 + 0.0877i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.12591 - 0.0494779i\)
\(L(\frac12)\) \(\approx\) \(1.12591 - 0.0494779i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-1.64e6 - 6.57e5i)T \)
good2 \( 1 + (92.8 + 30.1i)T + (3.31e3 + 2.40e3i)T^{2} \)
3 \( 1 + (-691. + 502. i)T + (1.64e5 - 5.05e5i)T^{2} \)
5 \( 1 + (-1.08e3 - 3.33e3i)T + (-1.97e8 + 1.43e8i)T^{2} \)
7 \( 1 + (6.92e4 - 9.52e4i)T + (-4.27e9 - 1.31e10i)T^{2} \)
13 \( 1 + (-4.52e6 - 1.47e6i)T + (1.88e13 + 1.36e13i)T^{2} \)
17 \( 1 + (7.78e6 - 2.52e6i)T + (4.71e14 - 3.42e14i)T^{2} \)
19 \( 1 + (1.08e6 + 1.49e6i)T + (-6.83e14 + 2.10e15i)T^{2} \)
23 \( 1 - 2.35e8T + 2.19e16T^{2} \)
29 \( 1 + (3.92e8 - 5.40e8i)T + (-1.09e17 - 3.36e17i)T^{2} \)
31 \( 1 + (-2.64e8 + 8.13e8i)T + (-6.37e17 - 4.62e17i)T^{2} \)
37 \( 1 + (1.37e9 + 9.99e8i)T + (2.03e18 + 6.26e18i)T^{2} \)
41 \( 1 + (-3.78e9 - 5.21e9i)T + (-6.97e18 + 2.14e19i)T^{2} \)
43 \( 1 - 2.12e9iT - 3.99e19T^{2} \)
47 \( 1 + (1.36e10 - 9.92e9i)T + (3.59e19 - 1.10e20i)T^{2} \)
53 \( 1 + (-9.66e9 + 2.97e10i)T + (-3.97e20 - 2.88e20i)T^{2} \)
59 \( 1 + (-5.24e10 - 3.81e10i)T + (5.49e20 + 1.69e21i)T^{2} \)
61 \( 1 + (2.08e9 - 6.76e8i)T + (2.14e21 - 1.56e21i)T^{2} \)
67 \( 1 + 7.27e10T + 8.18e21T^{2} \)
71 \( 1 + (-6.57e10 - 2.02e11i)T + (-1.32e22 + 9.64e21i)T^{2} \)
73 \( 1 + (-5.98e10 + 8.23e10i)T + (-7.07e21 - 2.17e22i)T^{2} \)
79 \( 1 + (1.35e11 + 4.38e10i)T + (4.78e22 + 3.47e22i)T^{2} \)
83 \( 1 + (-2.44e11 + 7.93e10i)T + (8.64e22 - 6.28e22i)T^{2} \)
89 \( 1 + 5.82e11T + 2.46e23T^{2} \)
97 \( 1 + (-3.57e11 + 1.10e12i)T + (-5.61e23 - 4.07e23i)T^{2} \)
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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

−18.03099646766086545888089430716, −16.46297647616280232890574112920, −14.63474137338457630479355053005, −12.93764329189102170818635829536, −11.21567449755606367626758591624, −9.342755184431857365755205235268, −8.542653844530522172090177493443, −6.87791563435361865899439025146, −2.78893782603982772855093620698, −1.39746042111428461150977304555, 0.897913211714376345609481315800, 3.63560201818387560797514407432, 6.81051774846980390878034773156, 8.586447616133713434404834373948, 9.389259056807474328104983928926, 10.73668427700965614624043563218, 13.49990353684071829019062685637, 15.14494169028651366874688779798, 16.32774626851183325695421684465, 17.32432714466063976788414744597

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