Properties

Degree 2
Conductor $ 2 \cdot 11 $
Sign $0.982 + 0.187i$
Motivic weight 3
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 + (2.64 − 8.12i)3-s + (1.23 + 3.80i)4-s + (−10.3 + 7.48i)5-s + (13.8 − 10.0i)6-s + (7.24 + 22.3i)7-s + (−2.47 + 7.60i)8-s + (−37.2 − 27.0i)9-s − 25.4·10-s + (−3.69 − 36.2i)11-s + 34.1·12-s + (−9.27 − 6.73i)13-s + (−14.4 + 44.6i)14-s + (33.6 + 103. i)15-s + (−12.9 + 9.40i)16-s + (52.9 − 38.5i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.508 − 1.56i)3-s + (0.154 + 0.475i)4-s + (−0.921 + 0.669i)5-s + (0.940 − 0.683i)6-s + (0.391 + 1.20i)7-s + (−0.109 + 0.336i)8-s + (−1.37 − 1.00i)9-s − 0.805·10-s + (−0.101 − 0.994i)11-s + 0.822·12-s + (−0.197 − 0.143i)13-s + (−0.276 + 0.851i)14-s + (0.578 + 1.78i)15-s + (−0.202 + 0.146i)16-s + (0.756 − 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22\)    =    \(2 \cdot 11\)
\( \varepsilon \)  =  $0.982 + 0.187i$
motivic weight  =  \(3\)
character  :  $\chi_{22} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 22,\ (\ :3/2),\ 0.982 + 0.187i)$
$L(2)$  $\approx$  $1.46851 - 0.139106i$
$L(\frac12)$  $\approx$  $1.46851 - 0.139106i$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-1.61 - 1.17i)T \)
11 \( 1 + (3.69 + 36.2i)T \)
good3 \( 1 + (-2.64 + 8.12i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (10.3 - 7.48i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-7.24 - 22.3i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (9.27 + 6.73i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-52.9 + 38.5i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (2.24 - 6.89i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 104.T + 1.21e4T^{2} \)
29 \( 1 + (39.3 + 121. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-233. - 169. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (26.3 + 81.2i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (41.8 - 128. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 353.T + 7.95e4T^{2} \)
47 \( 1 + (41.5 - 128. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (405. + 294. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-201. - 619. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (295. - 215. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 294.T + 3.00e5T^{2} \)
71 \( 1 + (107. - 77.8i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-145. - 446. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (330. + 239. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-1.09e3 + 799. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + 260.T + 7.04e5T^{2} \)
97 \( 1 + (1.14e3 + 831. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.84720920597659943333311155277, −15.89907952099138465069666035170, −14.70335721688984096085401876945, −13.77275429335399216464026987839, −12.29778520273756866983536395200, −11.59280781308044609981474328437, −8.419504813096985549378070924759, −7.54063715403432348854756224300, −6.01124755013911372474612167340, −2.87928150513178740951327685236, 3.90984967770940474486673319475, 4.68985297667322811981123857566, 7.931538838213455941251650101620, 9.726698556761715444130582992579, 10.74991367147899407513172985048, 12.23816710967871922496994421460, 13.99733360190087296919988055990, 15.04729612948247917659406567298, 16.01658358390686157830831115439, 17.11678320107260991577745086854

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