Properties

Degree 2
Conductor 41
Sign $0.752 + 0.658i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.746 − 1.02i)2-s + (−0.604 − 0.604i)3-s + (0.119 + 0.366i)4-s + (−3.27 + 1.06i)5-s + (−1.07 + 0.169i)6-s + (1.70 + 0.270i)7-s + (2.88 + 0.936i)8-s − 2.26i·9-s + (−1.35 + 4.16i)10-s + (−3.99 + 2.03i)11-s + (0.149 − 0.293i)12-s + (−0.380 − 2.40i)13-s + (1.55 − 1.55i)14-s + (2.62 + 1.33i)15-s + (2.49 − 1.81i)16-s + (−0.138 − 0.272i)17-s + ⋯
L(s)  = 1  + (0.528 − 0.726i)2-s + (−0.348 − 0.348i)3-s + (0.0596 + 0.183i)4-s + (−1.46 + 0.476i)5-s + (−0.437 + 0.0693i)6-s + (0.644 + 0.102i)7-s + (1.01 + 0.331i)8-s − 0.756i·9-s + (−0.428 + 1.31i)10-s + (−1.20 + 0.613i)11-s + (0.0432 − 0.0848i)12-s + (−0.105 − 0.665i)13-s + (0.414 − 0.414i)14-s + (0.678 + 0.345i)15-s + (0.622 − 0.452i)16-s + (−0.0336 − 0.0660i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.752 + 0.658i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.752 + 0.658i)$
$L(1)$  $\approx$  $0.774720 - 0.291262i$
$L(\frac12)$  $\approx$  $0.774720 - 0.291262i$
$L(\frac{3}{2})$   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 \( 1 + (5.99 + 2.26i)T \)
good2 \( 1 + (-0.746 + 1.02i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.604 + 0.604i)T + 3iT^{2} \)
5 \( 1 + (3.27 - 1.06i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.70 - 0.270i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (3.99 - 2.03i)T + (6.46 - 8.89i)T^{2} \)
13 \( 1 + (0.380 + 2.40i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (0.138 + 0.272i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.274 + 1.73i)T + (-18.0 - 5.87i)T^{2} \)
23 \( 1 + (-3.75 - 2.72i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-4.39 + 8.62i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (2.28 - 7.03i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.28 - 7.02i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (1.53 - 2.10i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (-6.25 + 0.991i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.556 - 1.09i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-2.66 - 1.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.655 + 0.901i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (9.09 + 4.63i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-0.475 + 0.242i)T + (41.7 - 57.4i)T^{2} \)
73 \( 1 + 8.56iT - 73T^{2} \)
79 \( 1 + (6.09 + 6.09i)T + 79iT^{2} \)
83 \( 1 - 9.88T + 83T^{2} \)
89 \( 1 + (14.4 + 2.28i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (-8.80 - 4.48i)T + (57.0 + 78.4i)T^{2} \)
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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

−15.68057684480751691658416938960, −15.00084224625442364492752742668, −13.28446288993087614240105169329, −12.18413766115363639184615165674, −11.57748128952741576715584642387, −10.52994717671362094264211576274, −8.098786230908710392721848436711, −7.21567512002102969431178447836, −4.77555394704808933441539587068, −3.14904209337022793535857170845, 4.39929894846140477493809162559, 5.33510224238542686136933842392, 7.35249842049865434798877876516, 8.319906534918270958501589424317, 10.59928834467512232288649242983, 11.36103112710109228297398216851, 12.91593862192748845772365410514, 14.22760617806750282344843790298, 15.33222237060705732308077072748, 16.19143183559549817402173216799

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