Properties

Label 2-11-11.8-c12-0-6
Degree $2$
Conductor $11$
Sign $-0.525 + 0.850i$
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  + (−0.697 − 0.960i)2-s + (−145. − 448. i)3-s + (1.26e3 − 3.89e3i)4-s + (1.88e4 + 1.36e4i)5-s + (−329. + 452. i)6-s + (−1.69e5 − 5.50e4i)7-s + (−9.24e3 + 3.00e3i)8-s + (2.49e5 − 1.81e5i)9-s − 2.76e4i·10-s + (−5.55e5 − 1.68e6i)11-s − 1.93e6·12-s + (−2.13e6 − 2.93e6i)13-s + (6.53e4 + 2.01e5i)14-s + (3.38e6 − 1.04e7i)15-s + (−1.35e7 − 9.85e6i)16-s + (2.26e6 − 3.11e6i)17-s + ⋯
L(s)  = 1  + (−0.0109 − 0.0150i)2-s + (−0.199 − 0.615i)3-s + (0.308 − 0.950i)4-s + (1.20 + 0.874i)5-s + (−0.00705 + 0.00970i)6-s + (−1.44 − 0.468i)7-s + (−0.0352 + 0.0114i)8-s + (0.470 − 0.341i)9-s − 0.0276i·10-s + (−0.313 − 0.949i)11-s − 0.646·12-s + (−0.441 − 0.607i)13-s + (0.00868 + 0.0267i)14-s + (0.297 − 0.915i)15-s + (−0.808 − 0.587i)16-s + (0.0938 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(10.0539\)
Root analytic conductor: \(3.17079\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :6),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.750032 - 1.34555i\)
\(L(\frac12)\) \(\approx\) \(0.750032 - 1.34555i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (5.55e5 + 1.68e6i)T \)
good2 \( 1 + (0.697 + 0.960i)T + (-1.26e3 + 3.89e3i)T^{2} \)
3 \( 1 + (145. + 448. i)T + (-4.29e5 + 3.12e5i)T^{2} \)
5 \( 1 + (-1.88e4 - 1.36e4i)T + (7.54e7 + 2.32e8i)T^{2} \)
7 \( 1 + (1.69e5 + 5.50e4i)T + (1.11e10 + 8.13e9i)T^{2} \)
13 \( 1 + (2.13e6 + 2.93e6i)T + (-7.19e12 + 2.21e13i)T^{2} \)
17 \( 1 + (-2.26e6 + 3.11e6i)T + (-1.80e14 - 5.54e14i)T^{2} \)
19 \( 1 + (6.69e7 - 2.17e7i)T + (1.79e15 - 1.30e15i)T^{2} \)
23 \( 1 - 1.53e8T + 2.19e16T^{2} \)
29 \( 1 + (-6.56e8 - 2.13e8i)T + (2.86e17 + 2.07e17i)T^{2} \)
31 \( 1 + (-1.22e9 + 8.88e8i)T + (2.43e17 - 7.49e17i)T^{2} \)
37 \( 1 + (5.93e8 - 1.82e9i)T + (-5.32e18 - 3.86e18i)T^{2} \)
41 \( 1 + (-4.26e9 + 1.38e9i)T + (1.82e19 - 1.32e19i)T^{2} \)
43 \( 1 + 5.46e9iT - 3.99e19T^{2} \)
47 \( 1 + (-3.96e9 - 1.22e10i)T + (-9.40e19 + 6.82e19i)T^{2} \)
53 \( 1 + (6.19e9 - 4.49e9i)T + (1.51e20 - 4.67e20i)T^{2} \)
59 \( 1 + (1.14e8 - 3.52e8i)T + (-1.43e21 - 1.04e21i)T^{2} \)
61 \( 1 + (-3.02e10 + 4.17e10i)T + (-8.20e20 - 2.52e21i)T^{2} \)
67 \( 1 - 7.40e9T + 8.18e21T^{2} \)
71 \( 1 + (-8.45e10 - 6.14e10i)T + (5.07e21 + 1.56e22i)T^{2} \)
73 \( 1 + (-7.29e10 - 2.36e10i)T + (1.85e22 + 1.34e22i)T^{2} \)
79 \( 1 + (1.02e10 + 1.41e10i)T + (-1.82e22 + 5.61e22i)T^{2} \)
83 \( 1 + (1.63e11 - 2.25e11i)T + (-3.30e22 - 1.01e23i)T^{2} \)
89 \( 1 + 1.07e11T + 2.46e23T^{2} \)
97 \( 1 + (-5.91e11 + 4.29e11i)T + (2.14e23 - 6.59e23i)T^{2} \)
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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

−17.21815364514696049427166020801, −15.50436334134421594212898719884, −13.97731575996221945310921322613, −12.89668839868875534770379022539, −10.59783509488035788915418545256, −9.789272404832876276007065799823, −6.71530314969649892426769594834, −6.06515139165986399366629437000, −2.66388131206825583910075423016, −0.72468959738178747314051707621, 2.38144195111618269968999708283, 4.64480734712972691828239391024, 6.63978323779219804202426336108, 9.019376608424447098857980715755, 10.12664368598288172318414538449, 12.50036610591393210205751276339, 13.15718845084980109584140698363, 15.61407419292690327743979193328, 16.60351381619909771518590373919, 17.49109951414927450033868918638

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