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  :  $\chi_{41} (39, \cdot )$
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

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

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