Properties

Degree 2
Conductor 11
Sign $-0.117 + 0.993i$
Motivic weight 4
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

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $-0.117 + 0.993i$
motivic weight  =  \(4\)
character  :  $\chi_{11} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :2),\ -0.117 + 0.993i)$
$L(\frac{5}{2})$  $\approx$  $0.563691 - 0.634285i$
$L(\frac12)$  $\approx$  $0.563691 - 0.634285i$
$L(3)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 11$, \(F_p\) is a polynomial of degree 2. If $p = 11$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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