Properties

Label 2-11-11.7-c4-0-0
Degree $2$
Conductor $11$
Sign $-0.117 - 0.993i$
Analytic cond. $1.13706$
Root an. cond. $1.06633$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.46 + 3.39i)2-s + (−2.26 + 6.96i)3-s + (−0.502 − 1.54i)4-s + (14.7 − 10.7i)5-s + (−18.0 − 24.8i)6-s + (47.9 − 15.5i)7-s + (−57.3 − 18.6i)8-s + (22.1 + 16.0i)9-s + 76.4i·10-s + (−82.9 + 88.1i)11-s + 11.9·12-s + (125. − 173. i)13-s + (−65.4 + 201. i)14-s + (41.2 + 126. i)15-s + (226. − 164. i)16-s + (−255. − 351. i)17-s + ⋯
L(s)  = 1  + (−0.616 + 0.849i)2-s + (−0.251 + 0.773i)3-s + (−0.0314 − 0.0966i)4-s + (0.589 − 0.428i)5-s + (−0.501 − 0.690i)6-s + (0.979 − 0.318i)7-s + (−0.896 − 0.291i)8-s + (0.273 + 0.198i)9-s + 0.764i·10-s + (−0.685 + 0.728i)11-s + 0.0827·12-s + (0.744 − 1.02i)13-s + (−0.334 + 1.02i)14-s + (0.183 + 0.563i)15-s + (0.882 − 0.641i)16-s + (−0.884 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $-0.117 - 0.993i$
Analytic conductor: \(1.13706\)
Root analytic conductor: \(1.06633\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :2),\ -0.117 - 0.993i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.563691 + 0.634285i\)
\(L(\frac12)\) \(\approx\) \(0.563691 + 0.634285i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (82.9 - 88.1i)T \)
good2 \( 1 + (2.46 - 3.39i)T + (-4.94 - 15.2i)T^{2} \)
3 \( 1 + (2.26 - 6.96i)T + (-65.5 - 47.6i)T^{2} \)
5 \( 1 + (-14.7 + 10.7i)T + (193. - 594. i)T^{2} \)
7 \( 1 + (-47.9 + 15.5i)T + (1.94e3 - 1.41e3i)T^{2} \)
13 \( 1 + (-125. + 173. i)T + (-8.82e3 - 2.71e4i)T^{2} \)
17 \( 1 + (255. + 351. i)T + (-2.58e4 + 7.94e4i)T^{2} \)
19 \( 1 + (61.3 + 19.9i)T + (1.05e5 + 7.66e4i)T^{2} \)
23 \( 1 - 536.T + 2.79e5T^{2} \)
29 \( 1 + (481. - 156. i)T + (5.72e5 - 4.15e5i)T^{2} \)
31 \( 1 + (-192. - 139. i)T + (2.85e5 + 8.78e5i)T^{2} \)
37 \( 1 + (718. + 2.21e3i)T + (-1.51e6 + 1.10e6i)T^{2} \)
41 \( 1 + (100. + 32.8i)T + (2.28e6 + 1.66e6i)T^{2} \)
43 \( 1 - 2.05e3iT - 3.41e6T^{2} \)
47 \( 1 + (-17.8 + 55.0i)T + (-3.94e6 - 2.86e6i)T^{2} \)
53 \( 1 + (-926. - 673. i)T + (2.43e6 + 7.50e6i)T^{2} \)
59 \( 1 + (596. + 1.83e3i)T + (-9.80e6 + 7.12e6i)T^{2} \)
61 \( 1 + (375. + 516. i)T + (-4.27e6 + 1.31e7i)T^{2} \)
67 \( 1 + 8.69e3T + 2.01e7T^{2} \)
71 \( 1 + (-1.51e3 + 1.10e3i)T + (7.85e6 - 2.41e7i)T^{2} \)
73 \( 1 + (-2.33e3 + 758. i)T + (2.29e7 - 1.66e7i)T^{2} \)
79 \( 1 + (2.25e3 - 3.10e3i)T + (-1.20e7 - 3.70e7i)T^{2} \)
83 \( 1 + (-4.35e3 - 5.98e3i)T + (-1.46e7 + 4.51e7i)T^{2} \)
89 \( 1 - 1.11e4T + 6.27e7T^{2} \)
97 \( 1 + (-9.94e3 - 7.22e3i)T + (2.73e7 + 8.41e7i)T^{2} \)
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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

−20.66507136637449232192066293871, −18.14076654266803196914957196275, −17.40877107680613737473068046071, −16.11646429932984226164139369236, −15.16017459602817783357528936197, −13.10763129324312994479317710392, −10.86069264021659937307262281901, −9.236559603220645234158218300419, −7.55148709967729826471510068919, −5.12814304516471770097483648457, 1.78430449216247841846235652431, 6.24415329828003939825929847300, 8.661434412446824451035715861938, 10.58873070303653872093683753306, 11.68938823081098180674161657375, 13.42558368241778379914659171738, 15.10554114173494619447216826701, 17.44399043857828401384921204341, 18.42430858504073478470348764815, 19.03317574022461201171228346690

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