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Properties

Label 2-11-11.10-c8-0-3
Degree $2$
Conductor $11$
Sign $0.332 - 0.943i$
Analytic cond. $4.48116$
Root an. cond. $2.11687$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.0i·2-s + 84.5·3-s + 59.3·4-s + 63.0·5-s + 1.18e3i·6-s + 1.66e3i·7-s + 4.42e3i·8-s + 582.·9-s + 884. i·10-s + (4.86e3 − 1.38e4i)11-s + 5.01e3·12-s − 1.61e4i·13-s − 2.33e4·14-s + 5.32e3·15-s − 4.68e4·16-s − 5.82e4i·17-s + ⋯
L(s)  = 1  + 0.876i·2-s + 1.04·3-s + 0.231·4-s + 0.100·5-s + 0.914i·6-s + 0.694i·7-s + 1.07i·8-s + 0.0887·9-s + 0.0884i·10-s + (0.332 − 0.943i)11-s + 0.241·12-s − 0.566i·13-s − 0.608·14-s + 0.105·15-s − 0.714·16-s − 0.697i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11\)
Sign: $0.332 - 0.943i$
Analytic conductor: \(4.48116\)
Root analytic conductor: \(2.11687\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 11,\ (\ :4),\ 0.332 - 0.943i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.78911 + 1.26656i\)
\(L(\frac12)\) \(\approx\) \(1.78911 + 1.26656i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-4.86e3 + 1.38e4i)T \)
good2 \( 1 - 14.0iT - 256T^{2} \)
3 \( 1 - 84.5T + 6.56e3T^{2} \)
5 \( 1 - 63.0T + 3.90e5T^{2} \)
7 \( 1 - 1.66e3iT - 5.76e6T^{2} \)
13 \( 1 + 1.61e4iT - 8.15e8T^{2} \)
17 \( 1 + 5.82e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.72e4iT - 1.69e10T^{2} \)
23 \( 1 - 2.90e5T + 7.83e10T^{2} \)
29 \( 1 + 1.15e6iT - 5.00e11T^{2} \)
31 \( 1 + 5.04e5T + 8.52e11T^{2} \)
37 \( 1 + 5.62e5T + 3.51e12T^{2} \)
41 \( 1 - 9.57e5iT - 7.98e12T^{2} \)
43 \( 1 - 6.28e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.14e6T + 2.38e13T^{2} \)
53 \( 1 + 5.44e6T + 6.22e13T^{2} \)
59 \( 1 + 1.63e7T + 1.46e14T^{2} \)
61 \( 1 - 1.66e7iT - 1.91e14T^{2} \)
67 \( 1 - 2.30e7T + 4.06e14T^{2} \)
71 \( 1 + 1.66e7T + 6.45e14T^{2} \)
73 \( 1 - 5.01e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.05e7iT - 1.51e15T^{2} \)
83 \( 1 + 4.65e7iT - 2.25e15T^{2} \)
89 \( 1 - 9.49e7T + 3.93e15T^{2} \)
97 \( 1 + 3.42e7T + 7.83e15T^{2} \)
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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

−19.00253606306714551156566277056, −17.27645964664228780005052606454, −15.80070137687546990254844183104, −14.81267030084517479038022361942, −13.63746380045899782518112603201, −11.47056105360012306615555929408, −9.053650980498135914826333118106, −7.82519504149597608241120338744, −5.85417590102816912254671156201, −2.72988702990547371793035530900, 1.86310637002698378178010381099, 3.69023494888069122645372678250, 7.18077094961160684637032025729, 9.236785279790319063992905918690, 10.71780601541548590953311467667, 12.41707877893598020109933809826, 13.90374732011962460092528261892, 15.25423954456883422903737431540, 17.05078024156280932306721358951, 18.97730020908900731616768106635

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