Properties

Label 2-22-11.9-c11-0-5
Degree $2$
Conductor $22$
Sign $0.966 - 0.258i$
Analytic cond. $16.9035$
Root an. cond. $4.11139$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.88 − 30.4i)2-s + (106. − 77.1i)3-s + (−828. − 601. i)4-s + (1.48e3 + 4.57e3i)5-s + (−1.29e3 − 3.99e3i)6-s + (9.58e3 + 6.96e3i)7-s + (−2.65e4 + 1.92e4i)8-s + (−4.94e4 + 1.52e5i)9-s + 1.54e5·10-s + (3.99e5 + 3.55e5i)11-s − 1.34e5·12-s + (−2.16e4 + 6.65e4i)13-s + (3.06e5 − 2.22e5i)14-s + (5.11e5 + 3.71e5i)15-s + (3.24e5 + 9.97e5i)16-s + (7.02e5 + 2.16e6i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.252 − 0.183i)3-s + (−0.404 − 0.293i)4-s + (0.212 + 0.655i)5-s + (−0.0681 − 0.209i)6-s + (0.215 + 0.156i)7-s + (−0.286 + 0.207i)8-s + (−0.278 + 0.858i)9-s + 0.487·10-s + (0.747 + 0.664i)11-s − 0.155·12-s + (−0.0161 + 0.0497i)13-s + (0.152 − 0.110i)14-s + (0.173 + 0.126i)15-s + (0.0772 + 0.237i)16-s + (0.119 + 0.369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.966 - 0.258i$
Analytic conductor: \(16.9035\)
Root analytic conductor: \(4.11139\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :11/2),\ 0.966 - 0.258i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.17958 + 0.286071i\)
\(L(\frac12)\) \(\approx\) \(2.17958 + 0.286071i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.88 + 30.4i)T \)
11 \( 1 + (-3.99e5 - 3.55e5i)T \)
good3 \( 1 + (-106. + 77.1i)T + (5.47e4 - 1.68e5i)T^{2} \)
5 \( 1 + (-1.48e3 - 4.57e3i)T + (-3.95e7 + 2.87e7i)T^{2} \)
7 \( 1 + (-9.58e3 - 6.96e3i)T + (6.11e8 + 1.88e9i)T^{2} \)
13 \( 1 + (2.16e4 - 6.65e4i)T + (-1.44e12 - 1.05e12i)T^{2} \)
17 \( 1 + (-7.02e5 - 2.16e6i)T + (-2.77e13 + 2.01e13i)T^{2} \)
19 \( 1 + (-1.67e6 + 1.21e6i)T + (3.59e13 - 1.10e14i)T^{2} \)
23 \( 1 - 5.38e7T + 9.52e14T^{2} \)
29 \( 1 + (-3.05e7 - 2.21e7i)T + (3.77e15 + 1.16e16i)T^{2} \)
31 \( 1 + (5.00e7 - 1.54e8i)T + (-2.05e16 - 1.49e16i)T^{2} \)
37 \( 1 + (1.36e8 + 9.95e7i)T + (5.49e16 + 1.69e17i)T^{2} \)
41 \( 1 + (6.10e8 - 4.43e8i)T + (1.70e17 - 5.23e17i)T^{2} \)
43 \( 1 + 1.16e9T + 9.29e17T^{2} \)
47 \( 1 + (5.46e8 - 3.96e8i)T + (7.63e17 - 2.35e18i)T^{2} \)
53 \( 1 + (9.63e8 - 2.96e9i)T + (-7.49e18 - 5.44e18i)T^{2} \)
59 \( 1 + (-4.78e9 - 3.47e9i)T + (9.31e18 + 2.86e19i)T^{2} \)
61 \( 1 + (-6.55e8 - 2.01e9i)T + (-3.52e19 + 2.55e19i)T^{2} \)
67 \( 1 - 1.14e10T + 1.22e20T^{2} \)
71 \( 1 + (3.54e9 + 1.09e10i)T + (-1.86e20 + 1.35e20i)T^{2} \)
73 \( 1 + (-1.00e10 - 7.29e9i)T + (9.69e19 + 2.98e20i)T^{2} \)
79 \( 1 + (2.56e9 - 7.90e9i)T + (-6.05e20 - 4.39e20i)T^{2} \)
83 \( 1 + (1.54e10 + 4.74e10i)T + (-1.04e21 + 7.56e20i)T^{2} \)
89 \( 1 + 1.85e10T + 2.77e21T^{2} \)
97 \( 1 + (-3.27e10 + 1.00e11i)T + (-5.78e21 - 4.20e21i)T^{2} \)
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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

−14.97336898746694693713490487974, −14.11570924281979093654042727628, −12.83917064720272020433809658476, −11.41054590259517692763124978895, −10.27952769636924292030893598699, −8.716883209533973712775085082696, −6.91334706359796733767231302242, −4.96226832403189890582725702878, −3.03257405931656070835332035876, −1.62068940816815527893074420967, 0.844485778366030246870327793111, 3.48029732157289989001976788353, 5.14107638411473838699975981227, 6.68926781200576371118319224854, 8.495396729091903804524684405083, 9.445291884061261498734038131900, 11.54029770137225176841403177664, 12.97709379450983544250781221943, 14.21828253528151948704367481653, 15.23269615890038095773995155267

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