Properties

Label 2-11-11.5-c9-0-3
Degree $2$
Conductor $11$
Sign $0.354 - 0.935i$
Analytic cond. $5.66539$
Root an. cond. $2.38020$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.40 + 13.5i)2-s + (169. + 122. i)3-s + (249. − 181. i)4-s + (63.1 − 194. i)5-s + (−920. + 2.83e3i)6-s + (−1.58e3 + 1.15e3i)7-s + (9.46e3 + 6.87e3i)8-s + (7.42e3 + 2.28e4i)9-s + 2.91e3·10-s + (−4.62e4 + 1.49e4i)11-s + 6.46e4·12-s + (−1.67e4 − 5.16e4i)13-s + (−2.25e4 − 1.64e4i)14-s + (3.45e4 − 2.51e4i)15-s + (−2.63e3 + 8.09e3i)16-s + (1.24e5 − 3.83e5i)17-s + ⋯
L(s)  = 1  + (0.194 + 0.598i)2-s + (1.20 + 0.876i)3-s + (0.488 − 0.354i)4-s + (0.0451 − 0.139i)5-s + (−0.290 + 0.892i)6-s + (−0.249 + 0.181i)7-s + (0.816 + 0.593i)8-s + (0.377 + 1.16i)9-s + 0.0921·10-s + (−0.951 + 0.306i)11-s + 0.899·12-s + (−0.162 − 0.501i)13-s + (−0.157 − 0.114i)14-s + (0.176 − 0.128i)15-s + (−0.0100 + 0.0308i)16-s + (0.362 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.354 - 0.935i$
Analytic conductor: \(5.66539\)
Root analytic conductor: \(2.38020\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :9/2),\ 0.354 - 0.935i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.17417 + 1.50128i\)
\(L(\frac12)\) \(\approx\) \(2.17417 + 1.50128i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (4.62e4 - 1.49e4i)T \)
good2 \( 1 + (-4.40 - 13.5i)T + (-414. + 300. i)T^{2} \)
3 \( 1 + (-169. - 122. i)T + (6.08e3 + 1.87e4i)T^{2} \)
5 \( 1 + (-63.1 + 194. i)T + (-1.58e6 - 1.14e6i)T^{2} \)
7 \( 1 + (1.58e3 - 1.15e3i)T + (1.24e7 - 3.83e7i)T^{2} \)
13 \( 1 + (1.67e4 + 5.16e4i)T + (-8.57e9 + 6.23e9i)T^{2} \)
17 \( 1 + (-1.24e5 + 3.83e5i)T + (-9.59e10 - 6.97e10i)T^{2} \)
19 \( 1 + (4.33e5 + 3.14e5i)T + (9.97e10 + 3.06e11i)T^{2} \)
23 \( 1 - 1.65e6T + 1.80e12T^{2} \)
29 \( 1 + (4.60e6 - 3.34e6i)T + (4.48e12 - 1.37e13i)T^{2} \)
31 \( 1 + (2.77e6 + 8.53e6i)T + (-2.13e13 + 1.55e13i)T^{2} \)
37 \( 1 + (3.75e6 - 2.73e6i)T + (4.01e13 - 1.23e14i)T^{2} \)
41 \( 1 + (6.01e6 + 4.36e6i)T + (1.01e14 + 3.11e14i)T^{2} \)
43 \( 1 + 2.24e7T + 5.02e14T^{2} \)
47 \( 1 + (-3.63e7 - 2.64e7i)T + (3.45e14 + 1.06e15i)T^{2} \)
53 \( 1 + (-5.04e6 - 1.55e7i)T + (-2.66e15 + 1.93e15i)T^{2} \)
59 \( 1 + (-3.55e7 + 2.58e7i)T + (2.67e15 - 8.23e15i)T^{2} \)
61 \( 1 + (4.25e7 - 1.30e8i)T + (-9.46e15 - 6.87e15i)T^{2} \)
67 \( 1 + 2.31e8T + 2.72e16T^{2} \)
71 \( 1 + (-5.56e7 + 1.71e8i)T + (-3.70e16 - 2.69e16i)T^{2} \)
73 \( 1 + (-5.65e7 + 4.10e7i)T + (1.81e16 - 5.59e16i)T^{2} \)
79 \( 1 + (-1.49e8 - 4.61e8i)T + (-9.69e16 + 7.04e16i)T^{2} \)
83 \( 1 + (-5.27e7 + 1.62e8i)T + (-1.51e17 - 1.09e17i)T^{2} \)
89 \( 1 - 3.08e8T + 3.50e17T^{2} \)
97 \( 1 + (-3.92e7 - 1.20e8i)T + (-6.15e17 + 4.46e17i)T^{2} \)
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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

−18.88663848090359386352849156320, −16.65302118639729355890989171437, −15.40613415060751819131676729150, −14.82934394025471181630174047448, −13.27393678362511741326794810964, −10.71871024758409645409399377882, −9.230309149002564037382312720865, −7.48643625609591347041551203470, −5.09155508649033647533579290150, −2.72565283510831581272714660628, 1.89658651694402792827196504236, 3.32177356130365272727492673682, 7.00856998814705610400837977469, 8.379525753719441216662588691171, 10.59082165803249876809143091298, 12.52474885153764305262175472352, 13.37692742088982714101608196050, 14.88882352893269136049501337061, 16.67503746964658028551550703882, 18.71451443935224171529378615045

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