Properties

Degree 2
Conductor 41
Sign $0.878 - 0.477i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.14 + 0.372i)2-s + (−1.55 + 1.55i)3-s + (−0.439 − 0.319i)4-s + (1.49 − 2.05i)5-s + (−2.35 + 1.20i)6-s + (−1.01 − 0.517i)7-s + (−1.80 − 2.48i)8-s − 1.81i·9-s + (2.48 − 1.80i)10-s + (−0.807 + 5.10i)11-s + (1.17 − 0.186i)12-s + (1.17 + 2.30i)13-s + (−0.972 − 0.972i)14-s + (0.873 + 5.51i)15-s + (−0.809 − 2.48i)16-s + (−3.05 − 0.483i)17-s + ⋯
L(s)  = 1  + (0.811 + 0.263i)2-s + (−0.895 + 0.895i)3-s + (−0.219 − 0.159i)4-s + (0.669 − 0.920i)5-s + (−0.963 + 0.490i)6-s + (−0.383 − 0.195i)7-s + (−0.637 − 0.878i)8-s − 0.604i·9-s + (0.785 − 0.570i)10-s + (−0.243 + 1.53i)11-s + (0.339 − 0.0538i)12-s + (0.325 + 0.638i)13-s + (−0.259 − 0.259i)14-s + (0.225 + 1.42i)15-s + (−0.202 − 0.622i)16-s + (−0.741 − 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.878 - 0.477i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (20, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.878 - 0.477i)$
$L(1)$  $\approx$  $0.825492 + 0.210038i$
$L(\frac12)$  $\approx$  $0.825492 + 0.210038i$
$L(\frac{3}{2})$   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 \( 1 + (5.78 - 2.74i)T \)
good2 \( 1 + (-1.14 - 0.372i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.55 - 1.55i)T - 3iT^{2} \)
5 \( 1 + (-1.49 + 2.05i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.01 + 0.517i)T + (4.11 + 5.66i)T^{2} \)
11 \( 1 + (0.807 - 5.10i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (-1.17 - 2.30i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (3.05 + 0.483i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-3.74 + 7.34i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (1.17 - 3.62i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.72 + 0.906i)T + (27.5 - 8.96i)T^{2} \)
31 \( 1 + (-3.65 + 2.65i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.584 - 0.424i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-1.75 - 0.570i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.15 - 1.60i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (5.52 - 0.875i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-0.242 + 0.747i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.70 + 2.17i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.835 + 5.27i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-0.708 + 4.47i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + 0.149iT - 73T^{2} \)
79 \( 1 + (11.4 - 11.4i)T - 79iT^{2} \)
83 \( 1 - 4.27T + 83T^{2} \)
89 \( 1 + (1.51 + 0.773i)T + (52.3 + 72.0i)T^{2} \)
97 \( 1 + (-0.687 - 4.34i)T + (-92.2 + 29.9i)T^{2} \)
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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

−15.94645061709321059611017684051, −15.38699486266946475375024969178, −13.71894385769131893449391304240, −12.97055865451466931669709864848, −11.62727022768673333376724878543, −9.928620237808733890171669587838, −9.333779806333156863954333938080, −6.60876366455649726077362269843, −5.11119541724835225565003035793, −4.52866666992866491083707694709, 3.13521749474630646733837386435, 5.68210104319064129359417959246, 6.38191465166827101091511314513, 8.350704769266418666608343741867, 10.41001840997871758352321581813, 11.58923434241141894994121672284, 12.60793984432918427194545190064, 13.59137642014451857348407028509, 14.35884159628012017491184814524, 16.12179094000411865343355634580

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