Properties

Degree 2
Conductor $ 2 \cdot 11 $
Sign $0.938 + 0.346i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.618 − 1.90i)2-s + (6.31 + 4.59i)3-s + (−3.23 + 2.35i)4-s + (4.60 − 14.1i)5-s + (4.82 − 14.8i)6-s + (−17.6 + 12.7i)7-s + (6.47 + 4.70i)8-s + (10.5 + 32.3i)9-s − 29.8·10-s + (−29.3 + 21.6i)11-s − 31.2·12-s + (−13.6 − 41.8i)13-s + (35.2 + 25.5i)14-s + (94.2 − 68.4i)15-s + (4.94 − 15.2i)16-s + (−7.69 + 23.6i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (1.21 + 0.883i)3-s + (−0.404 + 0.293i)4-s + (0.412 − 1.26i)5-s + (0.328 − 1.01i)6-s + (−0.950 + 0.690i)7-s + (0.286 + 0.207i)8-s + (0.389 + 1.19i)9-s − 0.943·10-s + (−0.805 + 0.592i)11-s − 0.751·12-s + (−0.290 − 0.893i)13-s + (0.672 + 0.488i)14-s + (1.62 − 1.17i)15-s + (0.0772 − 0.237i)16-s + (−0.109 + 0.338i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(22\)    =    \(2 \cdot 11\)
\( \varepsilon \)  =  $0.938 + 0.346i$
motivic weight  =  \(3\)
character  :  $\chi_{22} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 22,\ (\ :3/2),\ 0.938 + 0.346i)$
$L(2)$  $\approx$  $1.25316 - 0.223823i$
$L(\frac12)$  $\approx$  $1.25316 - 0.223823i$
$L(\frac{5}{2})$   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 \( 1 + (0.618 + 1.90i)T \)
11 \( 1 + (29.3 - 21.6i)T \)
good3 \( 1 + (-6.31 - 4.59i)T + (8.34 + 25.6i)T^{2} \)
5 \( 1 + (-4.60 + 14.1i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (17.6 - 12.7i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (13.6 + 41.8i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (7.69 - 23.6i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (17.7 + 12.9i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 - 177.T + 1.21e4T^{2} \)
29 \( 1 + (-120. + 87.8i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-23.2 - 71.4i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (179. - 130. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (-204. - 148. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 130.T + 7.95e4T^{2} \)
47 \( 1 + (403. + 293. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (-3.99 - 12.3i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (28.7 - 20.9i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-166. + 511. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + 519.T + 3.00e5T^{2} \)
71 \( 1 + (-24.2 + 74.6i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (925. - 672. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-238. - 734. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-166. + 510. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 - 667.T + 7.04e5T^{2} \)
97 \( 1 + (55.5 + 170. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−17.46134087853847445872303573992, −16.02686291020687616948115552690, −15.08159423927992692314490681406, −13.26476189031180444230864641523, −12.58936343751051277697512990458, −10.19962962534319173189819788189, −9.301423293277170306973970017950, −8.379656075122304239408693573735, −4.92848022681168137791735169484, −2.86097182176356573253175167642, 2.94895309570397671560633500049, 6.61386950415302146317304446656, 7.41364396910379173819144166997, 9.096992526413269370071210002297, 10.56308027481412713032616410693, 13.09331555818937609463907910344, 13.91073110098523408403647863326, 14.74765101211215472321035145887, 16.26743582993032312458990226401, 17.86077411455328569533687840822

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