Properties

Label 2-22-11.3-c11-0-1
Degree $2$
Conductor $22$
Sign $0.166 - 0.986i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−25.8 + 18.8i)2-s + (2.50 + 7.69i)3-s + (316. − 973. i)4-s + (−1.00e4 − 7.27e3i)5-s + (−209. − 152. i)6-s + (−8.89e3 + 2.73e4i)7-s + (1.01e4 + 3.11e4i)8-s + (1.43e5 − 1.04e5i)9-s + 3.96e5·10-s + (−4.79e5 − 2.35e5i)11-s + 8.28e3·12-s + (1.17e5 − 8.55e4i)13-s + (−2.84e5 − 8.76e5i)14-s + (3.09e4 − 9.52e4i)15-s + (−8.48e5 − 6.16e5i)16-s + (5.91e6 + 4.30e6i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.00594 + 0.0182i)3-s + (0.154 − 0.475i)4-s + (−1.43 − 1.04i)5-s + (−0.0110 − 0.00799i)6-s + (−0.200 + 0.615i)7-s + (0.109 + 0.336i)8-s + (0.808 − 0.587i)9-s + 1.25·10-s + (−0.897 − 0.441i)11-s + 0.00961·12-s + (0.0879 − 0.0639i)13-s + (−0.141 − 0.435i)14-s + (0.0105 − 0.0323i)15-s + (−0.202 − 0.146i)16-s + (1.01 + 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(22\)    =    \(2 \cdot 11\)
Sign: $0.166 - 0.986i$
Analytic conductor: \(16.9035\)
Root analytic conductor: \(4.11139\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{22} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 22,\ (\ :11/2),\ 0.166 - 0.986i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.535066 + 0.452438i\)
\(L(\frac12)\) \(\approx\) \(0.535066 + 0.452438i\)
\(L(\frac{13}{2})\) 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 \( 1 + (4.79e5 + 2.35e5i)T \)
good3 \( 1 + (-2.50 - 7.69i)T + (-1.43e5 + 1.04e5i)T^{2} \)
5 \( 1 + (1.00e4 + 7.27e3i)T + (1.50e7 + 4.64e7i)T^{2} \)
7 \( 1 + (8.89e3 - 2.73e4i)T + (-1.59e9 - 1.16e9i)T^{2} \)
13 \( 1 + (-1.17e5 + 8.55e4i)T + (5.53e11 - 1.70e12i)T^{2} \)
17 \( 1 + (-5.91e6 - 4.30e6i)T + (1.05e13 + 3.25e13i)T^{2} \)
19 \( 1 + (-4.87e6 - 1.50e7i)T + (-9.42e13 + 6.84e13i)T^{2} \)
23 \( 1 + 3.40e7T + 9.52e14T^{2} \)
29 \( 1 + (4.40e7 - 1.35e8i)T + (-9.87e15 - 7.17e15i)T^{2} \)
31 \( 1 + (-1.17e8 + 8.54e7i)T + (7.85e15 - 2.41e16i)T^{2} \)
37 \( 1 + (-1.83e7 + 5.63e7i)T + (-1.43e17 - 1.04e17i)T^{2} \)
41 \( 1 + (-2.58e8 - 7.95e8i)T + (-4.45e17 + 3.23e17i)T^{2} \)
43 \( 1 - 9.13e8T + 9.29e17T^{2} \)
47 \( 1 + (7.30e8 + 2.24e9i)T + (-2.00e18 + 1.45e18i)T^{2} \)
53 \( 1 + (-3.21e9 + 2.33e9i)T + (2.86e18 - 8.81e18i)T^{2} \)
59 \( 1 + (-1.16e8 + 3.58e8i)T + (-2.43e19 - 1.77e19i)T^{2} \)
61 \( 1 + (-5.49e9 - 3.99e9i)T + (1.34e19 + 4.13e19i)T^{2} \)
67 \( 1 + 1.62e10T + 1.22e20T^{2} \)
71 \( 1 + (-1.22e10 - 8.92e9i)T + (7.14e19 + 2.19e20i)T^{2} \)
73 \( 1 + (6.22e9 - 1.91e10i)T + (-2.53e20 - 1.84e20i)T^{2} \)
79 \( 1 + (1.69e10 - 1.23e10i)T + (2.31e20 - 7.11e20i)T^{2} \)
83 \( 1 + (2.06e10 + 1.50e10i)T + (3.97e20 + 1.22e21i)T^{2} \)
89 \( 1 - 3.02e10T + 2.77e21T^{2} \)
97 \( 1 + (5.94e10 - 4.31e10i)T + (2.21e21 - 6.80e21i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.95171035156787551199390217047, −14.89993873717018709960083690415, −12.75119149274106493705219090172, −11.88095548272073024525069977561, −10.05312189681811865312808543833, −8.513720259295068093146523322387, −7.67059228543021695470830348025, −5.61361954263600489704995136786, −3.81694392017181645305792503198, −1.05446141579936557394271725467, 0.41532745678552956012229838618, 2.75709385405113435183061091569, 4.25258641339858302511668161053, 7.23011736585939716725896960343, 7.76022824664155170114280630525, 10.00234815391087159326926231487, 10.94582231691863306628376486493, 12.13687952279657081520018340205, 13.72160141488294863363165042220, 15.47808897491971028637604552321

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