Properties

Degree 2
Conductor 41
Sign $-0.489 - 0.872i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 2.21i)2-s + (1.24 + 1.24i)3-s + (−1.70 − 5.23i)4-s + (−1.15 + 0.375i)5-s + (−4.75 + 0.753i)6-s + (1.19 + 0.189i)7-s + (9.13 + 2.96i)8-s + 0.0941i·9-s + (1.02 − 3.16i)10-s + (0.836 − 0.426i)11-s + (4.39 − 8.62i)12-s + (0.289 + 1.82i)13-s + (−2.34 + 2.34i)14-s + (−1.90 − 0.969i)15-s + (−12.3 + 8.98i)16-s + (−1.02 − 2.01i)17-s + ⋯
L(s)  = 1  + (−1.13 + 1.56i)2-s + (0.718 + 0.718i)3-s + (−0.850 − 2.61i)4-s + (−0.516 + 0.167i)5-s + (−1.94 + 0.307i)6-s + (0.451 + 0.0715i)7-s + (3.22 + 1.04i)8-s + 0.0313i·9-s + (0.324 − 0.999i)10-s + (0.252 − 0.128i)11-s + (1.26 − 2.49i)12-s + (0.0802 + 0.506i)13-s + (−0.626 + 0.626i)14-s + (−0.491 − 0.250i)15-s + (−3.09 + 2.24i)16-s + (−0.248 − 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $-0.489 - 0.872i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ -0.489 - 0.872i)$
$L(1)$  $\approx$  $0.272055 + 0.464598i$
$L(\frac12)$  $\approx$  $0.272055 + 0.464598i$
$L(\frac{3}{2})$   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 \( 1 + (-4.89 - 4.13i)T \)
good2 \( 1 + (1.61 - 2.21i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.24 - 1.24i)T + 3iT^{2} \)
5 \( 1 + (1.15 - 0.375i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.19 - 0.189i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (-0.836 + 0.426i)T + (6.46 - 8.89i)T^{2} \)
13 \( 1 + (-0.289 - 1.82i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (1.02 + 2.01i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.815 + 5.15i)T + (-18.0 - 5.87i)T^{2} \)
23 \( 1 + (5.31 + 3.86i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.689 - 1.35i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (0.515 - 1.58i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.97 - 6.06i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.714 + 0.982i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (5.36 - 0.849i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (4.56 - 8.95i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (6.23 + 4.52i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.982 - 1.35i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-13.1 - 6.68i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-2.77 + 1.41i)T + (41.7 - 57.4i)T^{2} \)
73 \( 1 + 0.596iT - 73T^{2} \)
79 \( 1 + (5.89 + 5.89i)T + 79iT^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + (-6.08 - 0.964i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (15.0 + 7.65i)T + (57.0 + 78.4i)T^{2} \)
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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

−16.24497332730710166356632846383, −15.55858428697906511757327271478, −14.70489309577474725263301015235, −13.91030712045174478060010767639, −11.24324846493893354648720106063, −9.776692837590448855972138989618, −8.911420429040030354950456136904, −7.894931926478431279323471508741, −6.52932312526660619031835495947, −4.60918941050621403651065942239, 1.91115962881095603746001628930, 3.79708842512108554983900580860, 7.79298909044953836633838839117, 8.173206698376730060135772989523, 9.641219432062798780502879386462, 10.93975444980216376041209332288, 12.06612983880110747250697065609, 12.94870414072639491323500077310, 14.14346566609883630783092065711, 16.16468170434835274478985687004

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