Properties

Label 2-11-11.3-c13-0-11
Degree $2$
Conductor $11$
Sign $-0.865 - 0.501i$
Analytic cond. $11.7954$
Root an. cond. $3.43444$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (80.7 − 58.6i)2-s + (−728. − 2.24e3i)3-s + (548. − 1.68e3i)4-s + (−1.29e4 − 9.38e3i)5-s + (−1.90e5 − 1.38e5i)6-s + (9.44e4 − 2.90e5i)7-s + (1.97e5 + 6.09e5i)8-s + (−3.21e6 + 2.33e6i)9-s − 1.59e6·10-s + (5.82e6 − 7.41e5i)11-s − 4.18e6·12-s + (−8.46e6 + 6.14e6i)13-s + (−9.42e6 − 2.90e7i)14-s + (−1.16e7 + 3.58e7i)15-s + (6.35e7 + 4.61e7i)16-s + (−1.20e8 − 8.73e7i)17-s + ⋯
L(s)  = 1  + (0.892 − 0.648i)2-s + (−0.577 − 1.77i)3-s + (0.0669 − 0.205i)4-s + (−0.369 − 0.268i)5-s + (−1.66 − 1.21i)6-s + (0.303 − 0.933i)7-s + (0.267 + 0.821i)8-s + (−2.01 + 1.46i)9-s − 0.503·10-s + (0.992 − 0.126i)11-s − 0.404·12-s + (−0.486 + 0.353i)13-s + (−0.334 − 1.03i)14-s + (−0.263 + 0.811i)15-s + (0.946 + 0.687i)16-s + (−1.20 − 0.877i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(11.7954\)
Root analytic conductor: \(3.43444\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :13/2),\ -0.865 - 0.501i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.420294 + 1.56250i\)
\(L(\frac12)\) \(\approx\) \(0.420294 + 1.56250i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-5.82e6 + 7.41e5i)T \)
good2 \( 1 + (-80.7 + 58.6i)T + (2.53e3 - 7.79e3i)T^{2} \)
3 \( 1 + (728. + 2.24e3i)T + (-1.28e6 + 9.37e5i)T^{2} \)
5 \( 1 + (1.29e4 + 9.38e3i)T + (3.77e8 + 1.16e9i)T^{2} \)
7 \( 1 + (-9.44e4 + 2.90e5i)T + (-7.83e10 - 5.69e10i)T^{2} \)
13 \( 1 + (8.46e6 - 6.14e6i)T + (9.35e13 - 2.88e14i)T^{2} \)
17 \( 1 + (1.20e8 + 8.73e7i)T + (3.06e15 + 9.41e15i)T^{2} \)
19 \( 1 + (2.64e7 + 8.12e7i)T + (-3.40e16 + 2.47e16i)T^{2} \)
23 \( 1 + 3.67e8T + 5.04e17T^{2} \)
29 \( 1 + (-1.35e9 + 4.17e9i)T + (-8.30e18 - 6.03e18i)T^{2} \)
31 \( 1 + (-2.90e9 + 2.11e9i)T + (7.54e18 - 2.32e19i)T^{2} \)
37 \( 1 + (1.51e9 - 4.65e9i)T + (-1.97e20 - 1.43e20i)T^{2} \)
41 \( 1 + (-9.54e9 - 2.93e10i)T + (-7.48e20 + 5.43e20i)T^{2} \)
43 \( 1 + 2.41e9T + 1.71e21T^{2} \)
47 \( 1 + (3.45e10 + 1.06e11i)T + (-4.41e21 + 3.20e21i)T^{2} \)
53 \( 1 + (5.08e10 - 3.69e10i)T + (8.04e21 - 2.47e22i)T^{2} \)
59 \( 1 + (-1.46e11 + 4.49e11i)T + (-8.49e22 - 6.17e22i)T^{2} \)
61 \( 1 + (-4.95e10 - 3.59e10i)T + (5.00e22 + 1.53e23i)T^{2} \)
67 \( 1 + 8.27e11T + 5.48e23T^{2} \)
71 \( 1 + (-1.52e12 - 1.10e12i)T + (3.60e23 + 1.10e24i)T^{2} \)
73 \( 1 + (-1.36e11 + 4.21e11i)T + (-1.35e24 - 9.82e23i)T^{2} \)
79 \( 1 + (-7.47e11 + 5.43e11i)T + (1.44e24 - 4.43e24i)T^{2} \)
83 \( 1 + (-4.19e12 - 3.04e12i)T + (2.74e24 + 8.43e24i)T^{2} \)
89 \( 1 - 3.21e12T + 2.19e25T^{2} \)
97 \( 1 + (-1.06e13 + 7.75e12i)T + (2.07e25 - 6.40e25i)T^{2} \)
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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

−16.90850601446990601529702128329, −14.04442652269223727286926580704, −13.35171494684074641344717793313, −11.98675495780852163232847038726, −11.38407185586203108425847459878, −8.044271900757009181202884749653, −6.62447786980391984849889318257, −4.50652588428969625499161364362, −2.17023055773188015653237774026, −0.57809334067833150810858814393, 3.75230081010075575852436298881, 4.92540538039856115839175045007, 6.18975078792270535732283176063, 9.102904288504929015984400147290, 10.62087113725591027968105447309, 12.09974496668603542277515441984, 14.54713351770718546237896893735, 15.18177624620031210867085580324, 16.07187798612494869255332466837, 17.48997701592951478305154030167

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