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Properties

Label 2-11-11.2-c2-0-0
Degree $2$
Conductor $11$
Sign $0.998 + 0.0475i$
Analytic cond. $0.299728$
Root an. cond. $0.547474$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.690 − 0.224i)2-s + (−1.11 + 0.812i)3-s + (−2.80 − 2.04i)4-s + (1.23 + 3.80i)5-s + (0.954 − 0.310i)6-s + (5.85 − 8.05i)7-s + (3.19 + 4.39i)8-s + (−2.19 + 6.74i)9-s − 2.90i·10-s + (−10.3 − 3.66i)11-s + 4.79·12-s + (−5 − 1.62i)13-s + (−5.85 + 4.25i)14-s + (−4.47 − 3.24i)15-s + (3.07 + 9.45i)16-s + (14.5 − 4.72i)17-s + ⋯
L(s)  = 1  + (−0.345 − 0.112i)2-s + (−0.372 + 0.270i)3-s + (−0.702 − 0.510i)4-s + (0.247 + 0.760i)5-s + (0.159 − 0.0517i)6-s + (0.836 − 1.15i)7-s + (0.398 + 0.549i)8-s + (−0.243 + 0.749i)9-s − 0.290i·10-s + (−0.942 − 0.333i)11-s + 0.399·12-s + (−0.384 − 0.124i)13-s + (−0.418 + 0.303i)14-s + (−0.298 − 0.216i)15-s + (0.192 + 0.591i)16-s + (0.854 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.998 + 0.0475i$
Analytic conductor: \(0.299728\)
Root analytic conductor: \(0.547474\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :1),\ 0.998 + 0.0475i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.537672 - 0.0127849i\)
\(L(\frac12)\) \(\approx\) \(0.537672 - 0.0127849i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (10.3 + 3.66i)T \)
good2 \( 1 + (0.690 + 0.224i)T + (3.23 + 2.35i)T^{2} \)
3 \( 1 + (1.11 - 0.812i)T + (2.78 - 8.55i)T^{2} \)
5 \( 1 + (-1.23 - 3.80i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (-5.85 + 8.05i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (5 + 1.62i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-14.5 + 4.72i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-1.21 - 1.67i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 2.76T + 529T^{2} \)
29 \( 1 + (16.7 - 22.9i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (2.20 - 6.77i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-32.5 - 23.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (41.2 + 56.7i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 23.0iT - 1.84e3T^{2} \)
47 \( 1 + (22.0 - 16.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (3.54 - 10.8i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-1.83 - 1.33i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-21.5 + 6.98i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 38.4T + 4.48e3T^{2} \)
71 \( 1 + (-23.5 - 72.5i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-60.4 + 83.2i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (3.74 + 1.21i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-79.1 + 25.7i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 123.T + 7.92e3T^{2} \)
97 \( 1 + (23.9 - 73.6i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−20.38870687963728353403409205420, −18.90431046176222009219819132324, −17.83487323144211353161470291866, −16.65630534190446683282561441679, −14.55693974802130800132633714662, −13.63415939960201137791630209978, −10.91802326774198080675652898897, −10.22897205180947550939390734720, −7.85159363374975635698652658666, −5.11354937382164602751352405843, 5.26043379360225753227016445339, 8.082194301962094467619559610771, 9.418783603637110004561332066081, 11.94778841066133950882930639167, 12.96619860766929219369018928180, 14.93121693169646358630650555256, 16.74217073052870468891295584722, 17.83570089460569816815216474038, 18.61569626643303310705499196481, 20.75584658215635000222586315288

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