Properties

Degree 2
Conductor 41
Sign $-0.350 - 0.936i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.29 − 0.746i)2-s + (−1.67 + 1.67i)3-s + (3.09 + 2.25i)4-s + (−1.68 + 2.32i)5-s + (5.09 − 2.59i)6-s + (−1.86 − 0.952i)7-s + (−2.60 − 3.57i)8-s − 2.60i·9-s + (5.61 − 4.07i)10-s + (0.0609 − 0.384i)11-s + (−8.95 + 1.41i)12-s + (1.56 + 3.08i)13-s + (3.58 + 3.58i)14-s + (−1.06 − 6.71i)15-s + (0.932 + 2.87i)16-s + (5.05 + 0.799i)17-s + ⋯
L(s)  = 1  + (−1.62 − 0.527i)2-s + (−0.966 + 0.966i)3-s + (1.54 + 1.12i)4-s + (−0.755 + 1.03i)5-s + (2.07 − 1.05i)6-s + (−0.706 − 0.360i)7-s + (−0.919 − 1.26i)8-s − 0.867i·9-s + (1.77 − 1.28i)10-s + (0.0183 − 0.115i)11-s + (−2.58 + 0.409i)12-s + (0.435 + 0.854i)13-s + (0.957 + 0.957i)14-s + (−0.274 − 1.73i)15-s + (0.233 + 0.717i)16-s + (1.22 + 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $-0.350 - 0.936i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (20, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ -0.350 - 0.936i)$
$L(1)$  $\approx$  $0.119833 + 0.172707i$
$L(\frac12)$  $\approx$  $0.119833 + 0.172707i$
$L(\frac{3}{2})$   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 \( 1 + (-1.91 - 6.11i)T \)
good2 \( 1 + (2.29 + 0.746i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.67 - 1.67i)T - 3iT^{2} \)
5 \( 1 + (1.68 - 2.32i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.86 + 0.952i)T + (4.11 + 5.66i)T^{2} \)
11 \( 1 + (-0.0609 + 0.384i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (-1.56 - 3.08i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-5.05 - 0.799i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (2.92 - 5.74i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (-0.410 + 1.26i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.589 + 0.0934i)T + (27.5 - 8.96i)T^{2} \)
31 \( 1 + (0.996 - 0.724i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.07 + 1.51i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-9.42 - 3.06i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.97 - 1.00i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (5.34 - 0.845i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-2.04 + 6.28i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.08 + 1.97i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.874 - 5.51i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (1.09 - 6.91i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 + (-4.23 + 4.23i)T - 79iT^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 + (-0.298 - 0.151i)T + (52.3 + 72.0i)T^{2} \)
97 \( 1 + (-2.25 - 14.2i)T + (-92.2 + 29.9i)T^{2} \)
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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

−16.38145244093029666972054608118, −16.22615783618600417846532984183, −14.61832237815401599692655579432, −12.15229235517112372378647663861, −11.12129046768284965827182955932, −10.48326398570275898145530968160, −9.599730616830127548378172877238, −7.87729962890632857424208757283, −6.43756936897035188965081086007, −3.64536165846230725817615291892, 0.67873017047084977318755823106, 5.69987673977583922532033961753, 7.02992824233600870990561938794, 8.134877694135806866026031756458, 9.298155541628278730794967188136, 10.86101476602560273860277427019, 12.11840108879941865049982980484, 12.95274638964899006067616912356, 15.43829737516821744875854075100, 16.19546962463973182211117224795

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