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Properties

Label 2-11-11.10-c16-0-7
Degree $2$
Conductor $11$
Sign $0.108 - 0.994i$
Analytic cond. $17.8556$
Root an. cond. $4.22560$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 469. i·2-s − 9.95e3·3-s − 1.55e5·4-s + 6.57e5·5-s − 4.67e6i·6-s − 5.55e5i·7-s − 4.20e7i·8-s + 5.59e7·9-s + 3.08e8i·10-s + (2.33e7 − 2.13e8i)11-s + 1.54e9·12-s − 1.96e8i·13-s + 2.60e8·14-s − 6.54e9·15-s + 9.57e9·16-s + 5.90e9i·17-s + ⋯
L(s)  = 1  + 1.83i·2-s − 1.51·3-s − 2.36·4-s + 1.68·5-s − 2.78i·6-s − 0.0963i·7-s − 2.50i·8-s + 1.30·9-s + 3.08i·10-s + (0.108 − 0.994i)11-s + 3.58·12-s − 0.240i·13-s + 0.176·14-s − 2.55·15-s + 2.22·16-s + 0.846i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.108 - 0.994i$
Analytic conductor: \(17.8556\)
Root analytic conductor: \(4.22560\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :8),\ 0.108 - 0.994i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.867532 + 0.777772i\)
\(L(\frac12)\) \(\approx\) \(0.867532 + 0.777772i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.33e7 + 2.13e8i)T \)
good2 \( 1 - 469. iT - 6.55e4T^{2} \)
3 \( 1 + 9.95e3T + 4.30e7T^{2} \)
5 \( 1 - 6.57e5T + 1.52e11T^{2} \)
7 \( 1 + 5.55e5iT - 3.32e13T^{2} \)
13 \( 1 + 1.96e8iT - 6.65e17T^{2} \)
17 \( 1 - 5.90e9iT - 4.86e19T^{2} \)
19 \( 1 + 2.75e10iT - 2.88e20T^{2} \)
23 \( 1 + 5.68e9T + 6.13e21T^{2} \)
29 \( 1 - 7.02e11iT - 2.50e23T^{2} \)
31 \( 1 - 7.44e11T + 7.27e23T^{2} \)
37 \( 1 - 8.52e11T + 1.23e25T^{2} \)
41 \( 1 + 7.20e12iT - 6.37e25T^{2} \)
43 \( 1 + 4.98e12iT - 1.36e26T^{2} \)
47 \( 1 - 2.29e13T + 5.66e26T^{2} \)
53 \( 1 + 7.83e13T + 3.87e27T^{2} \)
59 \( 1 - 2.80e13T + 2.15e28T^{2} \)
61 \( 1 + 1.66e14iT - 3.67e28T^{2} \)
67 \( 1 - 2.15e14T + 1.64e29T^{2} \)
71 \( 1 - 2.38e14T + 4.16e29T^{2} \)
73 \( 1 - 5.43e14iT - 6.50e29T^{2} \)
79 \( 1 + 2.00e15iT - 2.30e30T^{2} \)
83 \( 1 - 8.23e14iT - 5.07e30T^{2} \)
89 \( 1 - 1.42e15T + 1.54e31T^{2} \)
97 \( 1 - 1.19e16T + 6.14e31T^{2} \)
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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

−17.10215696050071710452607213007, −15.81447147331600827785333835054, −14.08301498791167884805311079244, −13.01332653746714274751710461246, −10.58598754965985427427674965152, −8.975750082222836820572028756727, −6.71665779733273921568639335721, −5.90215305524781681958082291356, −5.04593787785108757598183623582, −0.70439067619686684250587513652, 1.05389738880014021043745149206, 2.21017303511084827616008359829, 4.67065487004486363150956173206, 5.97224220856728505968728455786, 9.644963996026495866649034845604, 10.26709957052387323549602229225, 11.69238722260110619540710824512, 12.66831358316208413871106227144, 13.95580299071953867266874531825, 16.97360739909516439283897834855

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