Properties

Label 2-11-11.3-c9-0-7
Degree $2$
Conductor $11$
Sign $-0.651 + 0.758i$
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  + (24.5 − 17.8i)2-s + (−45.7 − 140. i)3-s + (127. − 391. i)4-s + (−601. − 437. i)5-s + (−3.64e3 − 2.64e3i)6-s + (430. − 1.32e3i)7-s + (937. + 2.88e3i)8-s + (−1.82e3 + 1.32e3i)9-s − 2.26e4·10-s + (2.76e3 − 4.84e4i)11-s − 6.10e4·12-s + (7.21e4 − 5.24e4i)13-s + (−1.30e4 − 4.02e4i)14-s + (−3.40e4 + 1.04e5i)15-s + (2.45e5 + 1.78e5i)16-s + (2.37e5 + 1.72e5i)17-s + ⋯
L(s)  = 1  + (1.08 − 0.789i)2-s + (−0.326 − 1.00i)3-s + (0.248 − 0.765i)4-s + (−0.430 − 0.312i)5-s + (−1.14 − 0.833i)6-s + (0.0677 − 0.208i)7-s + (0.0809 + 0.249i)8-s + (−0.0925 + 0.0672i)9-s − 0.715·10-s + (0.0569 − 0.998i)11-s − 0.849·12-s + (0.701 − 0.509i)13-s + (−0.0910 − 0.280i)14-s + (−0.173 + 0.534i)15-s + (0.936 + 0.680i)16-s + (0.690 + 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.651 + 0.758i$
Analytic conductor: \(5.66539\)
Root analytic conductor: \(2.38020\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :9/2),\ -0.651 + 0.758i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.965694 - 2.10094i\)
\(L(\frac12)\) \(\approx\) \(0.965694 - 2.10094i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.76e3 + 4.84e4i)T \)
good2 \( 1 + (-24.5 + 17.8i)T + (158. - 486. i)T^{2} \)
3 \( 1 + (45.7 + 140. i)T + (-1.59e4 + 1.15e4i)T^{2} \)
5 \( 1 + (601. + 437. i)T + (6.03e5 + 1.85e6i)T^{2} \)
7 \( 1 + (-430. + 1.32e3i)T + (-3.26e7 - 2.37e7i)T^{2} \)
13 \( 1 + (-7.21e4 + 5.24e4i)T + (3.27e9 - 1.00e10i)T^{2} \)
17 \( 1 + (-2.37e5 - 1.72e5i)T + (3.66e10 + 1.12e11i)T^{2} \)
19 \( 1 + (-1.89e5 - 5.82e5i)T + (-2.61e11 + 1.89e11i)T^{2} \)
23 \( 1 - 3.85e5T + 1.80e12T^{2} \)
29 \( 1 + (1.33e6 - 4.10e6i)T + (-1.17e13 - 8.52e12i)T^{2} \)
31 \( 1 + (4.15e6 - 3.01e6i)T + (8.17e12 - 2.51e13i)T^{2} \)
37 \( 1 + (-5.92e6 + 1.82e7i)T + (-1.05e14 - 7.63e13i)T^{2} \)
41 \( 1 + (-7.78e6 - 2.39e7i)T + (-2.64e14 + 1.92e14i)T^{2} \)
43 \( 1 + 2.90e7T + 5.02e14T^{2} \)
47 \( 1 + (-8.55e6 - 2.63e7i)T + (-9.05e14 + 6.57e14i)T^{2} \)
53 \( 1 + (2.56e7 - 1.86e7i)T + (1.01e15 - 3.13e15i)T^{2} \)
59 \( 1 + (-3.08e7 + 9.50e7i)T + (-7.00e15 - 5.09e15i)T^{2} \)
61 \( 1 + (-1.15e8 - 8.37e7i)T + (3.61e15 + 1.11e16i)T^{2} \)
67 \( 1 - 9.99e7T + 2.72e16T^{2} \)
71 \( 1 + (2.15e8 + 1.56e8i)T + (1.41e16 + 4.36e16i)T^{2} \)
73 \( 1 + (-3.82e7 + 1.17e8i)T + (-4.76e16 - 3.46e16i)T^{2} \)
79 \( 1 + (-2.34e8 + 1.70e8i)T + (3.70e16 - 1.13e17i)T^{2} \)
83 \( 1 + (-3.24e8 - 2.36e8i)T + (5.77e16 + 1.77e17i)T^{2} \)
89 \( 1 + 7.83e8T + 3.50e17T^{2} \)
97 \( 1 + (2.31e8 - 1.68e8i)T + (2.34e17 - 7.23e17i)T^{2} \)
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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

−18.14191274474203389163888997941, −16.42136945582985418120701007860, −14.35546132124014897659190052016, −13.05879406977573330721315742589, −12.22598827099745075987918845586, −10.93919511348661436595324769325, −8.012535617734525229505441948644, −5.80550279685105643765522183235, −3.61997539049426198359135613655, −1.22213059233694565986227611701, 3.93311747308671758024253141226, 5.23929842827147776960144031286, 7.15180678660246246889042559832, 9.765973129139572551250704880894, 11.55080124719829837425019072716, 13.37416225982331540249154747939, 14.98785138704888848428479750835, 15.60471072078790053340669847179, 16.80761397793965169345290026897, 18.78182017571645000385974512718

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