Properties

Degree 2
Conductor $ 2 \cdot 5 $
Sign $0.984 - 0.177i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8 − 8i)2-s + (82.7 + 82.7i)3-s − 128i·4-s + (−33.6 + 624. i)5-s + 1.32e3·6-s + (2.71e3 − 2.71e3i)7-s + (−1.02e3 − 1.02e3i)8-s + 7.14e3i·9-s + (4.72e3 + 5.26e3i)10-s − 2.09e4·11-s + (1.05e4 − 1.05e4i)12-s + (−8.09e3 − 8.09e3i)13-s − 4.34e4i·14-s + (−5.44e4 + 4.88e4i)15-s − 1.63e4·16-s + (−3.14e3 + 3.14e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (1.02 + 1.02i)3-s − 0.5i·4-s + (−0.0538 + 0.998i)5-s + 1.02·6-s + (1.13 − 1.13i)7-s + (−0.250 − 0.250i)8-s + 1.08i·9-s + (0.472 + 0.526i)10-s − 1.43·11-s + (0.511 − 0.511i)12-s + (−0.283 − 0.283i)13-s − 1.13i·14-s + (−1.07 + 0.965i)15-s − 0.250·16-s + (−0.0376 + 0.0376i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(9-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $0.984 - 0.177i$
motivic weight  =  \(8\)
character  :  $\chi_{10} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 10,\ (\ :4),\ 0.984 - 0.177i)$
$L(\frac{9}{2})$  $\approx$  $2.43627 + 0.217379i$
$L(\frac12)$  $\approx$  $2.43627 + 0.217379i$
$L(5)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (-8 + 8i)T \)
5 \( 1 + (33.6 - 624. i)T \)
good3 \( 1 + (-82.7 - 82.7i)T + 6.56e3iT^{2} \)
7 \( 1 + (-2.71e3 + 2.71e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 2.09e4T + 2.14e8T^{2} \)
13 \( 1 + (8.09e3 + 8.09e3i)T + 8.15e8iT^{2} \)
17 \( 1 + (3.14e3 - 3.14e3i)T - 6.97e9iT^{2} \)
19 \( 1 + 6.92e4iT - 1.69e10T^{2} \)
23 \( 1 + (-1.08e3 - 1.08e3i)T + 7.83e10iT^{2} \)
29 \( 1 + 1.20e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.26e5T + 8.52e11T^{2} \)
37 \( 1 + (1.17e6 - 1.17e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 1.74e6T + 7.98e12T^{2} \)
43 \( 1 + (-3.03e6 - 3.03e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (5.83e6 - 5.83e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-1.97e6 - 1.97e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 1.58e7iT - 1.46e14T^{2} \)
61 \( 1 + 2.51e6T + 1.91e14T^{2} \)
67 \( 1 + (-1.47e7 + 1.47e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 8.22e6T + 6.45e14T^{2} \)
73 \( 1 + (3.03e7 + 3.03e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 4.69e7iT - 1.51e15T^{2} \)
83 \( 1 + (-2.81e7 - 2.81e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 7.91e7iT - 3.93e15T^{2} \)
97 \( 1 + (-1.47e6 + 1.47e6i)T - 7.83e15iT^{2} \)
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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

−19.54042629232839871684162640187, −17.85750253085818060856408635700, −15.58624105706325791902748358787, −14.58615806482881108838494555425, −13.62620102160974074703749549077, −10.97466073515949477971240202201, −10.12407096360998248948078790266, −7.81725539377555679860902636651, −4.53669008119461396286520916113, −2.84166717444066003911303309946, 2.16456074157436581167961862165, 5.22986146145030304615417834263, 7.81246732371750747120041890220, 8.648420345736611399246963327099, 12.09895903757120815560541844909, 13.16067869683265866034452964177, 14.46442209357481424235818528492, 15.79063758485814265398196514834, 17.74287176166650237656429495845, 18.90421261364371334819173431548

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