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  :  $\chi_{41} (39, \cdot )$
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

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

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