Properties

Label 2-22-11.3-c13-0-12
Degree $2$
Conductor $22$
Sign $-0.563 - 0.826i$
Analytic cond. $23.5908$
Root an. cond. $4.85703$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (51.7 − 37.6i)2-s + (−417. − 1.28e3i)3-s + (1.26e3 − 3.89e3i)4-s + (−3.40e4 − 2.47e4i)5-s + (−6.99e4 − 5.08e4i)6-s + (9.66e4 − 2.97e5i)7-s + (−8.10e4 − 2.49e5i)8-s + (−1.88e5 + 1.36e5i)9-s − 2.69e6·10-s + (−5.59e6 + 1.77e6i)11-s − 5.53e6·12-s + (1.20e7 − 8.75e6i)13-s + (−6.18e6 − 1.90e7i)14-s + (−1.75e7 + 5.40e7i)15-s + (−1.35e7 − 9.86e6i)16-s + (−2.63e7 − 1.91e7i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.330 − 1.01i)3-s + (0.154 − 0.475i)4-s + (−0.973 − 0.707i)5-s + (−0.612 − 0.444i)6-s + (0.310 − 0.955i)7-s + (−0.109 − 0.336i)8-s + (−0.117 + 0.0856i)9-s − 0.850·10-s + (−0.953 + 0.302i)11-s − 0.535·12-s + (0.692 − 0.503i)13-s + (−0.219 − 0.675i)14-s + (−0.397 + 1.22i)15-s + (−0.202 − 0.146i)16-s + (−0.265 − 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $-0.563 - 0.826i$
Analytic conductor: \(23.5908\)
Root analytic conductor: \(4.85703\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :13/2),\ -0.563 - 0.826i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.302501783\)
\(L(\frac12)\) \(\approx\) \(1.302501783\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-51.7 + 37.6i)T \)
11 \( 1 + (5.59e6 - 1.77e6i)T \)
good3 \( 1 + (417. + 1.28e3i)T + (-1.28e6 + 9.37e5i)T^{2} \)
5 \( 1 + (3.40e4 + 2.47e4i)T + (3.77e8 + 1.16e9i)T^{2} \)
7 \( 1 + (-9.66e4 + 2.97e5i)T + (-7.83e10 - 5.69e10i)T^{2} \)
13 \( 1 + (-1.20e7 + 8.75e6i)T + (9.35e13 - 2.88e14i)T^{2} \)
17 \( 1 + (2.63e7 + 1.91e7i)T + (3.06e15 + 9.41e15i)T^{2} \)
19 \( 1 + (-8.48e7 - 2.61e8i)T + (-3.40e16 + 2.47e16i)T^{2} \)
23 \( 1 - 8.50e8T + 5.04e17T^{2} \)
29 \( 1 + (6.52e8 - 2.00e9i)T + (-8.30e18 - 6.03e18i)T^{2} \)
31 \( 1 + (1.22e9 - 8.88e8i)T + (7.54e18 - 2.32e19i)T^{2} \)
37 \( 1 + (6.59e9 - 2.03e10i)T + (-1.97e20 - 1.43e20i)T^{2} \)
41 \( 1 + (8.97e9 + 2.76e10i)T + (-7.48e20 + 5.43e20i)T^{2} \)
43 \( 1 + 1.35e10T + 1.71e21T^{2} \)
47 \( 1 + (2.85e10 + 8.78e10i)T + (-4.41e21 + 3.20e21i)T^{2} \)
53 \( 1 + (-2.17e11 + 1.58e11i)T + (8.04e21 - 2.47e22i)T^{2} \)
59 \( 1 + (-1.78e11 + 5.48e11i)T + (-8.49e22 - 6.17e22i)T^{2} \)
61 \( 1 + (8.56e10 + 6.21e10i)T + (5.00e22 + 1.53e23i)T^{2} \)
67 \( 1 + 5.93e11T + 5.48e23T^{2} \)
71 \( 1 + (1.20e12 + 8.75e11i)T + (3.60e23 + 1.10e24i)T^{2} \)
73 \( 1 + (5.64e10 - 1.73e11i)T + (-1.35e24 - 9.82e23i)T^{2} \)
79 \( 1 + (-7.82e11 + 5.68e11i)T + (1.44e24 - 4.43e24i)T^{2} \)
83 \( 1 + (-2.80e12 - 2.03e12i)T + (2.74e24 + 8.43e24i)T^{2} \)
89 \( 1 + 6.90e12T + 2.19e25T^{2} \)
97 \( 1 + (7.49e12 - 5.44e12i)T + (2.07e25 - 6.40e25i)T^{2} \)
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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

−13.56270504968694280728881261758, −12.75195059791210163108747829148, −11.76046001278660863379892027319, −10.46240663166991365504527575834, −8.141700062582283621643616022502, −7.03617044007673425140773301676, −5.15420540315745313174888907213, −3.67098740592828938517674135239, −1.43916735338802642914735815616, −0.40320938484483237518396604146, 2.87928330113735150741865551876, 4.30423798993993801957488089725, 5.54526387704955924610680562614, 7.32366226101433934287238510776, 8.889154779269063821232173425311, 10.86093484159086975160761436627, 11.55190734452309573927613874004, 13.28314784488024100657310115975, 15.06657161802098840394459636995, 15.46956529684797864526945696242

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