Properties

Label 2-22-11.7-c4-0-3
Degree $2$
Conductor $22$
Sign $0.0849 + 0.996i$
Analytic cond. $2.27413$
Root an. cond. $1.50802$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.66 − 2.28i)2-s + (3.26 − 10.0i)3-s + (−2.47 − 7.60i)4-s + (−24.6 + 17.8i)5-s + (−17.5 − 24.1i)6-s + (86.5 − 28.1i)7-s + (−21.5 − 6.99i)8-s + (−24.6 − 17.9i)9-s + 86.0i·10-s + (18.6 + 119. i)11-s − 84.4·12-s + (−13.1 + 18.1i)13-s + (79.5 − 244. i)14-s + (99.3 + 305. i)15-s + (−51.7 + 37.6i)16-s + (123. + 169. i)17-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (0.362 − 1.11i)3-s + (−0.154 − 0.475i)4-s + (−0.985 + 0.715i)5-s + (−0.487 − 0.671i)6-s + (1.76 − 0.573i)7-s + (−0.336 − 0.109i)8-s + (−0.304 − 0.221i)9-s + 0.860i·10-s + (0.153 + 0.988i)11-s − 0.586·12-s + (−0.0779 + 0.107i)13-s + (0.405 − 1.24i)14-s + (0.441 + 1.35i)15-s + (−0.202 + 0.146i)16-s + (0.425 + 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0849 + 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0849 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.0849 + 0.996i$
Analytic conductor: \(2.27413\)
Root analytic conductor: \(1.50802\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :2),\ 0.0849 + 0.996i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.21190 - 1.11301i\)
\(L(\frac12)\) \(\approx\) \(1.21190 - 1.11301i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.66 + 2.28i)T \)
11 \( 1 + (-18.6 - 119. i)T \)
good3 \( 1 + (-3.26 + 10.0i)T + (-65.5 - 47.6i)T^{2} \)
5 \( 1 + (24.6 - 17.8i)T + (193. - 594. i)T^{2} \)
7 \( 1 + (-86.5 + 28.1i)T + (1.94e3 - 1.41e3i)T^{2} \)
13 \( 1 + (13.1 - 18.1i)T + (-8.82e3 - 2.71e4i)T^{2} \)
17 \( 1 + (-123. - 169. i)T + (-2.58e4 + 7.94e4i)T^{2} \)
19 \( 1 + (483. + 157. i)T + (1.05e5 + 7.66e4i)T^{2} \)
23 \( 1 + 61.8T + 2.79e5T^{2} \)
29 \( 1 + (459. - 149. i)T + (5.72e5 - 4.15e5i)T^{2} \)
31 \( 1 + (465. + 338. i)T + (2.85e5 + 8.78e5i)T^{2} \)
37 \( 1 + (78.1 + 240. i)T + (-1.51e6 + 1.10e6i)T^{2} \)
41 \( 1 + (343. + 111. i)T + (2.28e6 + 1.66e6i)T^{2} \)
43 \( 1 - 2.64e3iT - 3.41e6T^{2} \)
47 \( 1 + (-375. + 1.15e3i)T + (-3.94e6 - 2.86e6i)T^{2} \)
53 \( 1 + (1.95e3 + 1.42e3i)T + (2.43e6 + 7.50e6i)T^{2} \)
59 \( 1 + (-1.68e3 - 5.17e3i)T + (-9.80e6 + 7.12e6i)T^{2} \)
61 \( 1 + (2.16e3 + 2.98e3i)T + (-4.27e6 + 1.31e7i)T^{2} \)
67 \( 1 - 7.74e3T + 2.01e7T^{2} \)
71 \( 1 + (2.98e3 - 2.16e3i)T + (7.85e6 - 2.41e7i)T^{2} \)
73 \( 1 + (2.71e3 - 882. i)T + (2.29e7 - 1.66e7i)T^{2} \)
79 \( 1 + (-2.47e3 + 3.40e3i)T + (-1.20e7 - 3.70e7i)T^{2} \)
83 \( 1 + (6.19e3 + 8.53e3i)T + (-1.46e7 + 4.51e7i)T^{2} \)
89 \( 1 - 6.11e3T + 6.27e7T^{2} \)
97 \( 1 + (1.56e3 + 1.13e3i)T + (2.73e7 + 8.41e7i)T^{2} \)
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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

−17.41124037640074183158747850397, −14.97865360834713692473768712104, −14.43248238924548637865184728149, −12.94984191774660762639777844619, −11.71974538780218791033603867549, −10.70756294861064475391297280540, −8.106657047757612693461689310219, −7.12905601372572824750764953950, −4.33449392873367146407264188241, −1.83144922761744357141495034490, 4.03434484492506444843614661199, 5.18620625857112749101609176906, 8.048206998193170893027069487940, 8.825335567863311206675616207574, 11.04539953348415060013810787869, 12.23647075172332677673515064623, 14.24605152440085431514565873159, 15.09997187944999007214571015185, 15.99118666317607468809602889238, 17.06956468046749872330695838116

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