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Properties

Label 2-22-11.10-c6-0-2
Degree $2$
Conductor $22$
Sign $0.322 - 0.946i$
Analytic cond. $5.06118$
Root an. cond. $2.24970$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·2-s + 16.9·3-s − 32.0·4-s + 175.·5-s + 96.0i·6-s + 412. i·7-s − 181. i·8-s − 440.·9-s + 994. i·10-s + (1.25e3 + 429. i)11-s − 543.·12-s + 1.85e3i·13-s − 2.33e3·14-s + 2.98e3·15-s + 1.02e3·16-s − 8.10e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.629·3-s − 0.500·4-s + 1.40·5-s + 0.444i·6-s + 1.20i·7-s − 0.353i·8-s − 0.604·9-s + 0.994i·10-s + (0.946 + 0.322i)11-s − 0.314·12-s + 0.843i·13-s − 0.849·14-s + 0.884·15-s + 0.250·16-s − 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.322 - 0.946i$
Analytic conductor: \(5.06118\)
Root analytic conductor: \(2.24970\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :3),\ 0.322 - 0.946i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.66724 + 1.19330i\)
\(L(\frac12)\) \(\approx\) \(1.66724 + 1.19330i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.65iT \)
11 \( 1 + (-1.25e3 - 429. i)T \)
good3 \( 1 - 16.9T + 729T^{2} \)
5 \( 1 - 175.T + 1.56e4T^{2} \)
7 \( 1 - 412. iT - 1.17e5T^{2} \)
13 \( 1 - 1.85e3iT - 4.82e6T^{2} \)
17 \( 1 + 8.10e3iT - 2.41e7T^{2} \)
19 \( 1 + 9.52e3iT - 4.70e7T^{2} \)
23 \( 1 - 219.T + 1.48e8T^{2} \)
29 \( 1 + 6.79e3iT - 5.94e8T^{2} \)
31 \( 1 - 6.23e3T + 8.87e8T^{2} \)
37 \( 1 + 5.93e4T + 2.56e9T^{2} \)
41 \( 1 - 3.33e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.31e5iT - 6.32e9T^{2} \)
47 \( 1 - 7.50e4T + 1.07e10T^{2} \)
53 \( 1 + 2.31e5T + 2.21e10T^{2} \)
59 \( 1 - 1.28e5T + 4.21e10T^{2} \)
61 \( 1 - 2.68e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.81e5T + 9.04e10T^{2} \)
71 \( 1 - 1.62e5T + 1.28e11T^{2} \)
73 \( 1 + 3.31e5iT - 1.51e11T^{2} \)
79 \( 1 - 9.42e4iT - 2.43e11T^{2} \)
83 \( 1 - 6.76e5iT - 3.26e11T^{2} \)
89 \( 1 - 2.88e5T + 4.96e11T^{2} \)
97 \( 1 + 1.47e6T + 8.32e11T^{2} \)
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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

−17.03945852276760110069605638316, −15.51116005675675767605943074367, −14.26573630006693214783538448558, −13.63116283622699022085621697995, −11.81662145240111271099676312935, −9.349026904171294264455660232162, −8.991742616975132500305818912624, −6.71049817479345139679322113036, −5.29626245506729951983954756978, −2.40066914510890635042382070490, 1.55874727887623103869284834703, 3.57430447490615509589377678655, 5.98620159989822230140865238319, 8.320557258802263216212350192508, 9.781906085583886225253568861315, 10.75696897887652347940790332060, 12.74325910423787045390781953885, 13.90526660461465639756858610100, 14.49022822067732222420998003570, 17.00330377084142631821637932405

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