Properties

Degree 2
Conductor 11
Sign $0.844 - 0.534i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.02 + 1.41i)2-s + (2.68 + 8.25i)3-s + (4.00 − 12.3i)4-s + (−8.06 − 5.85i)5-s + (−8.89 + 12.2i)6-s + (−56.0 − 18.2i)7-s + (48.0 − 15.5i)8-s + (4.60 − 3.34i)9-s − 17.3i·10-s + (−16.1 + 119. i)11-s + 112.·12-s + (75.0 + 103. i)13-s + (−31.7 − 97.6i)14-s + (26.7 − 82.2i)15-s + (−96.6 − 70.1i)16-s + (22.8 − 31.4i)17-s + ⋯
L(s)  = 1  + (0.256 + 0.352i)2-s + (0.297 + 0.917i)3-s + (0.250 − 0.770i)4-s + (−0.322 − 0.234i)5-s + (−0.246 + 0.339i)6-s + (−1.14 − 0.371i)7-s + (0.750 − 0.243i)8-s + (0.0568 − 0.0412i)9-s − 0.173i·10-s + (−0.133 + 0.991i)11-s + 0.781·12-s + (0.444 + 0.611i)13-s + (−0.161 − 0.498i)14-s + (0.118 − 0.365i)15-s + (−0.377 − 0.274i)16-s + (0.0790 − 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11\)
\( \varepsilon \)  =  $0.844 - 0.534i$
motivic weight  =  \(4\)
character  :  $\chi_{11} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 11,\ (\ :2),\ 0.844 - 0.534i)$
$L(\frac{5}{2})$  $\approx$  $1.18135 + 0.342456i$
$L(\frac12)$  $\approx$  $1.18135 + 0.342456i$
$L(3)$   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 \( 1 + (16.1 - 119. i)T \)
good2 \( 1 + (-1.02 - 1.41i)T + (-4.94 + 15.2i)T^{2} \)
3 \( 1 + (-2.68 - 8.25i)T + (-65.5 + 47.6i)T^{2} \)
5 \( 1 + (8.06 + 5.85i)T + (193. + 594. i)T^{2} \)
7 \( 1 + (56.0 + 18.2i)T + (1.94e3 + 1.41e3i)T^{2} \)
13 \( 1 + (-75.0 - 103. i)T + (-8.82e3 + 2.71e4i)T^{2} \)
17 \( 1 + (-22.8 + 31.4i)T + (-2.58e4 - 7.94e4i)T^{2} \)
19 \( 1 + (560. - 182. i)T + (1.05e5 - 7.66e4i)T^{2} \)
23 \( 1 - 973.T + 2.79e5T^{2} \)
29 \( 1 + (141. + 46.0i)T + (5.72e5 + 4.15e5i)T^{2} \)
31 \( 1 + (-140. + 102. i)T + (2.85e5 - 8.78e5i)T^{2} \)
37 \( 1 + (-442. + 1.36e3i)T + (-1.51e6 - 1.10e6i)T^{2} \)
41 \( 1 + (561. - 182. i)T + (2.28e6 - 1.66e6i)T^{2} \)
43 \( 1 - 1.17e3iT - 3.41e6T^{2} \)
47 \( 1 + (151. + 465. i)T + (-3.94e6 + 2.86e6i)T^{2} \)
53 \( 1 + (1.80e3 - 1.31e3i)T + (2.43e6 - 7.50e6i)T^{2} \)
59 \( 1 + (1.48e3 - 4.58e3i)T + (-9.80e6 - 7.12e6i)T^{2} \)
61 \( 1 + (-2.83e3 + 3.90e3i)T + (-4.27e6 - 1.31e7i)T^{2} \)
67 \( 1 - 187.T + 2.01e7T^{2} \)
71 \( 1 + (537. + 390. i)T + (7.85e6 + 2.41e7i)T^{2} \)
73 \( 1 + (-5.02e3 - 1.63e3i)T + (2.29e7 + 1.66e7i)T^{2} \)
79 \( 1 + (5.01e3 + 6.90e3i)T + (-1.20e7 + 3.70e7i)T^{2} \)
83 \( 1 + (136. - 187. i)T + (-1.46e7 - 4.51e7i)T^{2} \)
89 \( 1 + 6.64e3T + 6.27e7T^{2} \)
97 \( 1 + (5.91e3 - 4.29e3i)T + (2.73e7 - 8.41e7i)T^{2} \)
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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

−20.02621226478315407667460885660, −18.95793137659585725997738433466, −16.62756924985976010394693466098, −15.65094696498285305620470359071, −14.69419195063749924195016914148, −12.90122949381567115469815998797, −10.56565454183933163404492163689, −9.416578457041041045029976374748, −6.70588737493782200969618209835, −4.34097181921691043144127527094, 3.06720870789742373251382605435, 6.79519065506185683550935256931, 8.419517639833572874400981910865, 11.03277751230077122785601969880, 12.76176450493755462113213728412, 13.29227715448769113774337395939, 15.55513645699437981220831877192, 16.96916956745881516936708924623, 18.83542575268301966417944163363, 19.45913143690613832663684433886

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