Properties

Label 2-22-11.3-c7-0-3
Degree $2$
Conductor $22$
Sign $0.817 + 0.575i$
Analytic cond. $6.87247$
Root an. cond. $2.62153$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.47 + 4.70i)2-s + (3.49 + 10.7i)3-s + (19.7 − 60.8i)4-s + (−150. − 109. i)5-s + (−73.1 − 53.1i)6-s + (47.2 − 145. i)7-s + (158. + 486. i)8-s + (1.66e3 − 1.21e3i)9-s + 1.48e3·10-s + (3.46e3 − 2.72e3i)11-s + 723.·12-s + (7.76e3 − 5.64e3i)13-s + (378. + 1.16e3i)14-s + (647. − 1.99e3i)15-s + (−3.31e3 − 2.40e3i)16-s + (488. + 355. i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.0746 + 0.229i)3-s + (0.154 − 0.475i)4-s + (−0.536 − 0.390i)5-s + (−0.138 − 0.100i)6-s + (0.0520 − 0.160i)7-s + (0.109 + 0.336i)8-s + (0.761 − 0.553i)9-s + 0.469·10-s + (0.785 − 0.618i)11-s + 0.120·12-s + (0.980 − 0.712i)13-s + (0.0368 + 0.113i)14-s + (0.0495 − 0.152i)15-s + (−0.202 − 0.146i)16-s + (0.0241 + 0.0175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(6.87247\)
Root analytic conductor: \(2.62153\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :7/2),\ 0.817 + 0.575i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.10942 - 0.351577i\)
\(L(\frac12)\) \(\approx\) \(1.10942 - 0.351577i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.47 - 4.70i)T \)
11 \( 1 + (-3.46e3 + 2.72e3i)T \)
good3 \( 1 + (-3.49 - 10.7i)T + (-1.76e3 + 1.28e3i)T^{2} \)
5 \( 1 + (150. + 109. i)T + (2.41e4 + 7.43e4i)T^{2} \)
7 \( 1 + (-47.2 + 145. i)T + (-6.66e5 - 4.84e5i)T^{2} \)
13 \( 1 + (-7.76e3 + 5.64e3i)T + (1.93e7 - 5.96e7i)T^{2} \)
17 \( 1 + (-488. - 355. i)T + (1.26e8 + 3.90e8i)T^{2} \)
19 \( 1 + (3.68e3 + 1.13e4i)T + (-7.23e8 + 5.25e8i)T^{2} \)
23 \( 1 + 7.30e4T + 3.40e9T^{2} \)
29 \( 1 + (192. - 591. i)T + (-1.39e10 - 1.01e10i)T^{2} \)
31 \( 1 + (-8.12e4 + 5.90e4i)T + (8.50e9 - 2.61e10i)T^{2} \)
37 \( 1 + (-1.99e4 + 6.13e4i)T + (-7.68e10 - 5.57e10i)T^{2} \)
41 \( 1 + (1.33e5 + 4.09e5i)T + (-1.57e11 + 1.14e11i)T^{2} \)
43 \( 1 - 3.69e5T + 2.71e11T^{2} \)
47 \( 1 + (-2.04e5 - 6.29e5i)T + (-4.09e11 + 2.97e11i)T^{2} \)
53 \( 1 + (1.31e6 - 9.52e5i)T + (3.63e11 - 1.11e12i)T^{2} \)
59 \( 1 + (2.19e5 - 6.75e5i)T + (-2.01e12 - 1.46e12i)T^{2} \)
61 \( 1 + (4.22e5 + 3.07e5i)T + (9.71e11 + 2.98e12i)T^{2} \)
67 \( 1 + 1.67e5T + 6.06e12T^{2} \)
71 \( 1 + (-4.22e6 - 3.07e6i)T + (2.81e12 + 8.64e12i)T^{2} \)
73 \( 1 + (-1.68e6 + 5.17e6i)T + (-8.93e12 - 6.49e12i)T^{2} \)
79 \( 1 + (-5.35e6 + 3.88e6i)T + (5.93e12 - 1.82e13i)T^{2} \)
83 \( 1 + (-2.44e6 - 1.77e6i)T + (8.38e12 + 2.58e13i)T^{2} \)
89 \( 1 + 1.10e7T + 4.42e13T^{2} \)
97 \( 1 + (6.22e6 - 4.52e6i)T + (2.49e13 - 7.68e13i)T^{2} \)
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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

−16.12961364495631722878216636373, −15.44375544719421167219370132719, −13.90585401933191153460694980216, −12.23022477664312327744580965674, −10.71492848217103543935102689618, −9.217073609155853623870679480146, −7.961080536392715309190158982699, −6.23132182312862729968139263533, −4.01564080783179611658209502699, −0.854667717212688796529406444565, 1.69594881240732968381740448838, 3.98558585414131858563147245595, 6.78368238724356196948640855987, 8.163832250960899990890924081220, 9.771447562810052815255953901658, 11.19808012535228877618185145768, 12.33641648476847977958153420922, 13.86428644119419217265422646624, 15.45647323333787165962064335867, 16.59788463162209376346946524037

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