Properties

Degree 2
Conductor 41
Sign $0.995 - 0.0985i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.05 + 0.666i)2-s + (1.49 − 1.49i)3-s + (2.14 − 1.55i)4-s + (1.72 + 2.37i)5-s + (−2.07 + 4.07i)6-s + (−1.11 − 2.18i)7-s + (−0.828 + 1.13i)8-s − 1.49i·9-s + (−5.12 − 3.72i)10-s + (−6.07 + 0.961i)11-s + (0.879 − 5.55i)12-s + (0.531 + 0.270i)13-s + (3.74 + 3.74i)14-s + (6.14 + 0.973i)15-s + (−0.700 + 2.15i)16-s + (0.0921 + 0.581i)17-s + ⋯
L(s)  = 1  + (−1.45 + 0.471i)2-s + (0.865 − 0.865i)3-s + (1.07 − 0.779i)4-s + (0.771 + 1.06i)5-s + (−0.847 + 1.66i)6-s + (−0.420 − 0.825i)7-s + (−0.292 + 0.402i)8-s − 0.498i·9-s + (−1.61 − 1.17i)10-s + (−1.83 + 0.289i)11-s + (0.254 − 1.60i)12-s + (0.147 + 0.0750i)13-s + (0.999 + 0.999i)14-s + (1.58 + 0.251i)15-s + (−0.175 + 0.539i)16-s + (0.0223 + 0.141i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.995 - 0.0985i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.995 - 0.0985i)$
$L(1)$  $\approx$  $0.538403 + 0.0265865i$
$L(\frac12)$  $\approx$  $0.538403 + 0.0265865i$
$L(\frac{3}{2})$   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 \( 1 + (1.29 - 6.27i)T \)
good2 \( 1 + (2.05 - 0.666i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.49 + 1.49i)T - 3iT^{2} \)
5 \( 1 + (-1.72 - 2.37i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.11 + 2.18i)T + (-4.11 + 5.66i)T^{2} \)
11 \( 1 + (6.07 - 0.961i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (-0.531 - 0.270i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.0921 - 0.581i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.67 - 0.854i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (1.21 + 3.75i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.785 + 4.96i)T + (-27.5 - 8.96i)T^{2} \)
31 \( 1 + (-4.03 - 2.93i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.87 - 1.36i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-2.50 + 0.812i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-2.33 + 4.57i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.818 + 5.16i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-3.23 - 9.94i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.968 - 0.314i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.42 + 0.542i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-4.74 + 0.751i)T + (67.5 - 21.9i)T^{2} \)
73 \( 1 + 0.596iT - 73T^{2} \)
79 \( 1 + (-3.13 + 3.13i)T - 79iT^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 + (5.50 + 10.8i)T + (-52.3 + 72.0i)T^{2} \)
97 \( 1 + (10.0 + 1.58i)T + (92.2 + 29.9i)T^{2} \)
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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

−16.40826269009497880178803293437, −15.13694263560768053226876451279, −13.81168016489242912067332332216, −13.06826395792872954161340913102, −10.51220248583086473207160577111, −10.10565938395711526172224858269, −8.374394779892348577191111967036, −7.45644436237480244544111947585, −6.51125133158008343000244526169, −2.44489912598634372225288123875, 2.60491280530989260121374932466, 5.31782159207568487687027912728, 8.108837522104956910841261828882, 8.972528813440242632518044321733, 9.660676228630893738252197966682, 10.66884950043157772632746824003, 12.49359305955094598734979107982, 13.72352310491304639199677893932, 15.54538717845470244223146438468, 16.12377638418059019955010433818

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