Properties

Degree 2
Conductor 41
Sign $0.435 + 0.900i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.522 − 0.718i)2-s + (−0.983 − 0.983i)3-s + (0.374 − 1.15i)4-s + (1.02 + 0.331i)5-s + (−0.193 + 1.22i)6-s + (0.625 + 3.95i)7-s + (−2.71 + 0.881i)8-s − 1.06i·9-s + (−0.294 − 0.906i)10-s + (−1.14 + 2.23i)11-s + (−1.50 + 0.764i)12-s + (4.25 + 0.673i)13-s + (2.51 − 2.51i)14-s + (−0.677 − 1.33i)15-s + (0.0903 + 0.0656i)16-s + (−4.26 − 2.17i)17-s + ⋯
L(s)  = 1  + (−0.369 − 0.508i)2-s + (−0.567 − 0.567i)3-s + (0.187 − 0.575i)4-s + (0.456 + 0.148i)5-s + (−0.0789 + 0.498i)6-s + (0.236 + 1.49i)7-s + (−0.959 + 0.311i)8-s − 0.355i·9-s + (−0.0931 − 0.286i)10-s + (−0.344 + 0.675i)11-s + (−0.433 + 0.220i)12-s + (1.17 + 0.186i)13-s + (0.671 − 0.671i)14-s + (−0.174 − 0.343i)15-s + (0.0225 + 0.0164i)16-s + (−1.03 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.435 + 0.900i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.435 + 0.900i)$
$L(1)$  $\approx$  $0.529805 - 0.332328i$
$L(\frac12)$  $\approx$  $0.529805 - 0.332328i$
$L(\frac{3}{2})$   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 \( 1 + (-6.18 - 1.67i)T \)
good2 \( 1 + (0.522 + 0.718i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.983 + 0.983i)T + 3iT^{2} \)
5 \( 1 + (-1.02 - 0.331i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.625 - 3.95i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (1.14 - 2.23i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (-4.25 - 0.673i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (4.26 + 2.17i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.967 + 0.153i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (1.78 - 1.29i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.83 + 0.933i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (2.02 + 6.23i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.34 - 10.3i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (1.18 + 1.63i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.55 + 9.83i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-6.91 + 3.52i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-7.77 + 5.64i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.263 - 0.362i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.59 + 3.12i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (6.39 - 12.5i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 + 8.75iT - 73T^{2} \)
79 \( 1 + (1.93 + 1.93i)T + 79iT^{2} \)
83 \( 1 + 2.79T + 83T^{2} \)
89 \( 1 + (-1.73 - 10.9i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (-1.70 - 3.35i)T + (-57.0 + 78.4i)T^{2} \)
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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

−15.72800896035446652497775567764, −14.99610721823153740436072419968, −13.40296242848478154593901218751, −11.97879548444413297695615244817, −11.38719044396598245395713878408, −9.841027988768288923414021811368, −8.745864650790782906789193249792, −6.50164936457221681323676186161, −5.57877161238320930343356587702, −2.12528537002233437980815050805, 3.99963340873282751388752589139, 5.93997108563676850315588121840, 7.46201995702442092921348755099, 8.744058179256188164361265142471, 10.49003596142143006122392595745, 11.16635018469040504977209631459, 13.03082603102690553601111444433, 13.94556186349786799466486619661, 15.85435968209232239552301554630, 16.31456743824355521337625524977

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