Properties

Degree 2
Conductor 41
Sign $0.860 - 0.509i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.415 + 0.571i)2-s + (−0.242 − 0.242i)3-s + (0.463 + 1.42i)4-s + (2.26 − 0.734i)5-s + (0.239 − 0.0379i)6-s + (−4.85 − 0.768i)7-s + (−2.35 − 0.764i)8-s − 2.88i·9-s + (−0.519 + 1.59i)10-s + (1.51 − 0.773i)11-s + (0.233 − 0.458i)12-s + (0.621 + 3.92i)13-s + (2.45 − 2.45i)14-s + (−0.727 − 0.370i)15-s + (−1.01 + 0.736i)16-s + (−1.24 − 2.44i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.404i)2-s + (−0.140 − 0.140i)3-s + (0.231 + 0.713i)4-s + (1.01 − 0.328i)5-s + (0.0977 − 0.0154i)6-s + (−1.83 − 0.290i)7-s + (−0.831 − 0.270i)8-s − 0.960i·9-s + (−0.164 + 0.505i)10-s + (0.457 − 0.233i)11-s + (0.0674 − 0.132i)12-s + (0.172 + 1.08i)13-s + (0.656 − 0.656i)14-s + (−0.187 − 0.0956i)15-s + (−0.253 + 0.184i)16-s + (−0.302 − 0.593i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.860 - 0.509i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.860 - 0.509i)$
$L(1)$  $\approx$  $0.665572 + 0.182124i$
$L(\frac12)$  $\approx$  $0.665572 + 0.182124i$
$L(\frac{3}{2})$   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 \( 1 + (4.01 - 4.98i)T \)
good2 \( 1 + (0.415 - 0.571i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.242 + 0.242i)T + 3iT^{2} \)
5 \( 1 + (-2.26 + 0.734i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (4.85 + 0.768i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (-1.51 + 0.773i)T + (6.46 - 8.89i)T^{2} \)
13 \( 1 + (-0.621 - 3.92i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (1.24 + 2.44i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.150 - 0.953i)T + (-18.0 - 5.87i)T^{2} \)
23 \( 1 + (-5.46 - 3.97i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.230 + 0.451i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (-0.182 + 0.561i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.31 - 4.04i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-3.16 + 4.35i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (4.96 - 0.786i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.46 + 6.79i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (6.81 + 4.95i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.408 - 0.562i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.63 + 1.85i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-6.47 + 3.30i)T + (41.7 - 57.4i)T^{2} \)
73 \( 1 + 9.72iT - 73T^{2} \)
79 \( 1 + (-6.15 - 6.15i)T + 79iT^{2} \)
83 \( 1 - 4.81T + 83T^{2} \)
89 \( 1 + (3.63 + 0.576i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (3.63 + 1.85i)T + (57.0 + 78.4i)T^{2} \)
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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

−16.57392055056131638640268127885, −15.43807764168878273598863522423, −13.64760379386703945321221802730, −12.84530482004556861044420399985, −11.70604148944424425202675642548, −9.561684119628150221536945073665, −9.140593853241567874471345478519, −6.87740214720251234944392900422, −6.25922175481662633268252364742, −3.41914894908407211552846829199, 2.65731920454566601151662690035, 5.65279933730860543172643679249, 6.61119379315011818088962725011, 9.095818168763041564548458480343, 10.10859765911058443011217053793, 10.74746689426050923850391207252, 12.61040323250584356160685938173, 13.57696058889594645801531636505, 14.99339687944176390386634466767, 16.03392353877640178895128574038

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