Properties

Label 2-22-11.4-c3-0-0
Degree $2$
Conductor $22$
Sign $0.0694 - 0.997i$
Analytic cond. $1.29804$
Root an. cond. $1.13931$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 − 1.17i)2-s + (−3.04 + 9.37i)3-s + (1.23 + 3.80i)4-s + (1.30 − 0.951i)5-s + (15.9 − 11.5i)6-s + (4.57 + 14.0i)7-s + (2.47 − 7.60i)8-s + (−56.7 − 41.2i)9-s − 3.23·10-s + (35.5 − 7.99i)11-s − 39.4·12-s + (41.1 + 29.9i)13-s + (9.14 − 28.1i)14-s + (4.92 + 15.1i)15-s + (−12.9 + 9.40i)16-s + (48.7 − 35.4i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.586 + 1.80i)3-s + (0.154 + 0.475i)4-s + (0.117 − 0.0850i)5-s + (1.08 − 0.788i)6-s + (0.246 + 0.759i)7-s + (0.109 − 0.336i)8-s + (−2.10 − 1.52i)9-s − 0.102·10-s + (0.975 − 0.219i)11-s − 0.948·12-s + (0.878 + 0.638i)13-s + (0.174 − 0.537i)14-s + (0.0848 + 0.261i)15-s + (−0.202 + 0.146i)16-s + (0.696 − 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(1.29804\)
Root analytic conductor: \(1.13931\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :3/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.543695 + 0.507145i\)
\(L(\frac12)\) \(\approx\) \(0.543695 + 0.507145i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.61 + 1.17i)T \)
11 \( 1 + (-35.5 + 7.99i)T \)
good3 \( 1 + (3.04 - 9.37i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (-1.30 + 0.951i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-4.57 - 14.0i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (-41.1 - 29.9i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-48.7 + 35.4i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (8.16 - 25.1i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 29.1T + 1.21e4T^{2} \)
29 \( 1 + (46.4 + 142. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (23.7 + 17.2i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (69.4 + 213. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-36.9 + 113. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 213.T + 7.95e4T^{2} \)
47 \( 1 + (54.0 - 166. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-51.1 - 37.1i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (80.9 + 249. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-573. + 416. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 151.T + 3.00e5T^{2} \)
71 \( 1 + (853. - 620. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (9.74 + 29.9i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (52.6 + 38.2i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-186. + 135. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + 618.T + 7.04e5T^{2} \)
97 \( 1 + (-889. - 646. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−17.54711040795928208514253046078, −16.57372336826201576297104095579, −15.68424085478847796279638277759, −14.37665718757283807328225623726, −11.88397381502875296852872429535, −11.11046832442640062795435444615, −9.656526450348949311601438725277, −8.846824525044530794240294505285, −5.77692110461709407696167239516, −3.86481701959384562163662199456, 1.21742615757564394572398109215, 6.00880382143101634036294082962, 7.15034933082561905444553428926, 8.348982683636203967668500566687, 10.67772981168577718714218575810, 11.99724676293981293732439633659, 13.36090016500174947128266615355, 14.43425228110415364384607791989, 16.56140133071725943875160461871, 17.48889833517301111251903578414

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