Properties

Label 2-22-11.4-c11-0-0
Degree $2$
Conductor $22$
Sign $-0.907 - 0.420i$
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  + (−25.8 − 18.8i)2-s + (−88.4 + 272. i)3-s + (316. + 973. i)4-s + (−354. + 257. i)5-s + (7.40e3 − 5.38e3i)6-s + (1.23e4 + 3.78e4i)7-s + (1.01e4 − 3.11e4i)8-s + (7.70e4 + 5.60e4i)9-s + 1.40e4·10-s + (7.66e4 − 5.28e5i)11-s − 2.92e5·12-s + (−3.03e5 − 2.20e5i)13-s + (3.93e5 − 1.21e6i)14-s + (−3.87e4 − 1.19e5i)15-s + (−8.48e5 + 6.16e5i)16-s + (−4.83e6 + 3.51e6i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.210 + 0.646i)3-s + (0.154 + 0.475i)4-s + (−0.0507 + 0.0368i)5-s + (0.388 − 0.282i)6-s + (0.276 + 0.851i)7-s + (0.109 − 0.336i)8-s + (0.435 + 0.316i)9-s + 0.0443·10-s + (0.143 − 0.989i)11-s − 0.339·12-s + (−0.226 − 0.164i)13-s + (0.195 − 0.602i)14-s + (−0.0131 − 0.0405i)15-s + (−0.202 + 0.146i)16-s + (−0.826 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $-0.907 - 0.420i$
Analytic conductor: \(16.9035\)
Root analytic conductor: \(4.11139\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :11/2),\ -0.907 - 0.420i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.121366 + 0.549878i\)
\(L(\frac12)\) \(\approx\) \(0.121366 + 0.549878i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (25.8 + 18.8i)T \)
11 \( 1 + (-7.66e4 + 5.28e5i)T \)
good3 \( 1 + (88.4 - 272. i)T + (-1.43e5 - 1.04e5i)T^{2} \)
5 \( 1 + (354. - 257. i)T + (1.50e7 - 4.64e7i)T^{2} \)
7 \( 1 + (-1.23e4 - 3.78e4i)T + (-1.59e9 + 1.16e9i)T^{2} \)
13 \( 1 + (3.03e5 + 2.20e5i)T + (5.53e11 + 1.70e12i)T^{2} \)
17 \( 1 + (4.83e6 - 3.51e6i)T + (1.05e13 - 3.25e13i)T^{2} \)
19 \( 1 + (3.86e6 - 1.19e7i)T + (-9.42e13 - 6.84e13i)T^{2} \)
23 \( 1 + 2.09e7T + 9.52e14T^{2} \)
29 \( 1 + (3.69e7 + 1.13e8i)T + (-9.87e15 + 7.17e15i)T^{2} \)
31 \( 1 + (1.81e8 + 1.32e8i)T + (7.85e15 + 2.41e16i)T^{2} \)
37 \( 1 + (6.87e6 + 2.11e7i)T + (-1.43e17 + 1.04e17i)T^{2} \)
41 \( 1 + (-2.52e7 + 7.77e7i)T + (-4.45e17 - 3.23e17i)T^{2} \)
43 \( 1 - 2.17e7T + 9.29e17T^{2} \)
47 \( 1 + (2.30e8 - 7.09e8i)T + (-2.00e18 - 1.45e18i)T^{2} \)
53 \( 1 + (1.53e9 + 1.11e9i)T + (2.86e18 + 8.81e18i)T^{2} \)
59 \( 1 + (-2.00e8 - 6.15e8i)T + (-2.43e19 + 1.77e19i)T^{2} \)
61 \( 1 + (-2.11e8 + 1.53e8i)T + (1.34e19 - 4.13e19i)T^{2} \)
67 \( 1 + 1.86e10T + 1.22e20T^{2} \)
71 \( 1 + (1.66e10 - 1.20e10i)T + (7.14e19 - 2.19e20i)T^{2} \)
73 \( 1 + (-6.06e9 - 1.86e10i)T + (-2.53e20 + 1.84e20i)T^{2} \)
79 \( 1 + (-2.85e10 - 2.07e10i)T + (2.31e20 + 7.11e20i)T^{2} \)
83 \( 1 + (-3.73e10 + 2.71e10i)T + (3.97e20 - 1.22e21i)T^{2} \)
89 \( 1 - 4.74e10T + 2.77e21T^{2} \)
97 \( 1 + (8.06e10 + 5.85e10i)T + (2.21e21 + 6.80e21i)T^{2} \)
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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

−16.06088527004298607822656116234, −14.95302494753206361253307527509, −13.12925373662950451716289903283, −11.65027783093525078678701812334, −10.59836530198411637804011590279, −9.277701379597682166103996969384, −7.958096544745408303369716344894, −5.78353102295504063995645884232, −3.90513344624750082369322285049, −1.94632675360472848836926019198, 0.26138745581047077450542518630, 1.77343080661188453513821059238, 4.53913480784054345639147622463, 6.70114247260302708118050851431, 7.46687266724657476720820494582, 9.249356484642107283084990390971, 10.66858577352753875284940610944, 12.19410815796030489726712932270, 13.56803414160202626721373037775, 14.95549690508104010978484897901

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