Properties

Degree 2
Conductor 41
Sign $0.994 + 0.104i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.698 − 0.227i)2-s + (0.0432 − 0.0432i)3-s + (−1.18 + 0.858i)4-s + (−0.422 − 0.581i)5-s + (0.0204 − 0.0400i)6-s + (−1.42 − 2.79i)7-s + (−1.49 + 2.05i)8-s + 2.99i·9-s + (−0.426 − 0.310i)10-s + (3.06 − 0.486i)11-s + (−0.0139 + 0.0883i)12-s + (−1.79 − 0.912i)13-s + (−1.62 − 1.62i)14-s + (−0.0434 − 0.00687i)15-s + (0.325 − 1.00i)16-s + (0.304 + 1.92i)17-s + ⋯
L(s)  = 1  + (0.494 − 0.160i)2-s + (0.0249 − 0.0249i)3-s + (−0.590 + 0.429i)4-s + (−0.188 − 0.259i)5-s + (0.00833 − 0.0163i)6-s + (−0.537 − 1.05i)7-s + (−0.528 + 0.727i)8-s + 0.998i·9-s + (−0.134 − 0.0980i)10-s + (0.925 − 0.146i)11-s + (−0.00403 + 0.0254i)12-s + (−0.496 − 0.253i)13-s + (−0.434 − 0.434i)14-s + (−0.0112 − 0.00177i)15-s + (0.0813 − 0.250i)16-s + (0.0738 + 0.466i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.994 + 0.104i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.994 + 0.104i)$
$L(1)$  $\approx$  $0.835041 - 0.0436022i$
$L(\frac12)$  $\approx$  $0.835041 - 0.0436022i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 41$,\(F_p(T)\) is a polynomial of degree 2. If $p = 41$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad41 \( 1 + (2.53 - 5.88i)T \)
good2 \( 1 + (-0.698 + 0.227i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.0432 + 0.0432i)T - 3iT^{2} \)
5 \( 1 + (0.422 + 0.581i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.42 + 2.79i)T + (-4.11 + 5.66i)T^{2} \)
11 \( 1 + (-3.06 + 0.486i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (1.79 + 0.912i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.304 - 1.92i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-3.91 + 1.99i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (-0.275 - 0.848i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.43 - 9.03i)T + (-27.5 - 8.96i)T^{2} \)
31 \( 1 + (5.64 + 4.10i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.49 - 2.53i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (10.3 - 3.35i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-5.24 + 10.2i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.465 + 2.94i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-1.37 - 4.23i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.3 - 3.69i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.00 - 1.26i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (0.988 - 0.156i)T + (67.5 - 21.9i)T^{2} \)
73 \( 1 + 6.49iT - 73T^{2} \)
79 \( 1 + (4.43 - 4.43i)T - 79iT^{2} \)
83 \( 1 - 7.83T + 83T^{2} \)
89 \( 1 + (6.40 + 12.5i)T + (-52.3 + 72.0i)T^{2} \)
97 \( 1 + (-6.34 - 1.00i)T + (92.2 + 29.9i)T^{2} \)
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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

−16.45835626911218201509334541324, −14.62033334032451171601941743585, −13.63583780197161964440837758094, −12.85197213835531834456984739462, −11.55447024735887503749312255694, −10.06338013601090994715979606888, −8.584769799172099122491399488848, −7.18910840401283579983722550119, −5.03634743241277028996081305016, −3.60681902096579331110948601170, 3.63319350333662729485157751021, 5.51888062903332874706047919324, 6.78576236485734024867186874343, 9.057669183344745945332560103058, 9.693081714650288816138822320267, 11.78075329035360643205292152430, 12.58398797726872987664783704029, 14.07543239404372128069644458617, 14.93045676364368917801659645847, 15.75557896136827528575591072269

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