Properties

Degree 2
Conductor $ 2 \cdot 5 $
Sign $0.916 + 0.400i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s + (1 + i)3-s − 8i·4-s + (−15 + 20i)5-s + 4·6-s + (−19 + 19i)7-s + (−16 − 16i)8-s − 79i·9-s + (10 + 70i)10-s + 202·11-s + (8 − 8i)12-s + (−99 − 99i)13-s + 76i·14-s + (−35 + 5i)15-s − 64·16-s + (−239 + 239i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.111 + 0.111i)3-s − 0.5i·4-s + (−0.599 + 0.800i)5-s + 0.111·6-s + (−0.387 + 0.387i)7-s + (−0.250 − 0.250i)8-s − 0.975i·9-s + (0.100 + 0.700i)10-s + 1.66·11-s + (0.0555 − 0.0555i)12-s + (−0.585 − 0.585i)13-s + 0.387i·14-s + (−0.155 + 0.0222i)15-s − 0.250·16-s + (−0.826 + 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $0.916 + 0.400i$
motivic weight  =  \(4\)
character  :  $\chi_{10} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 10,\ (\ :2),\ 0.916 + 0.400i)$
$L(\frac{5}{2})$  $\approx$  $1.19384 - 0.249270i$
$L(\frac12)$  $\approx$  $1.19384 - 0.249270i$
$L(3)$   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 \( 1 + (-2 + 2i)T \)
5 \( 1 + (15 - 20i)T \)
good3 \( 1 + (-1 - i)T + 81iT^{2} \)
7 \( 1 + (19 - 19i)T - 2.40e3iT^{2} \)
11 \( 1 - 202T + 1.46e4T^{2} \)
13 \( 1 + (99 + 99i)T + 2.85e4iT^{2} \)
17 \( 1 + (239 - 239i)T - 8.35e4iT^{2} \)
19 \( 1 + 40iT - 1.30e5T^{2} \)
23 \( 1 + (-541 - 541i)T + 2.79e5iT^{2} \)
29 \( 1 - 200iT - 7.07e5T^{2} \)
31 \( 1 + 758T + 9.23e5T^{2} \)
37 \( 1 + (-141 + 141i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.04e3T + 2.82e6T^{2} \)
43 \( 1 + (759 + 759i)T + 3.41e6iT^{2} \)
47 \( 1 + (459 - 459i)T - 4.87e6iT^{2} \)
53 \( 1 + (1.81e3 + 1.81e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 4.60e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.08e3T + 1.38e7T^{2} \)
67 \( 1 + (-5.08e3 + 5.08e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 3.47e3T + 2.54e7T^{2} \)
73 \( 1 + (3.47e3 + 3.47e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 7.68e3iT - 3.89e7T^{2} \)
83 \( 1 + (-6.08e3 - 6.08e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 5.68e3iT - 6.27e7T^{2} \)
97 \( 1 + (-561 + 561i)T - 8.85e7iT^{2} \)
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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

−20.01403317711729005848201304095, −19.20368663275779165457972978862, −17.55502990608387111124039039755, −15.35825312872810959797410897070, −14.54725242579615035720868542985, −12.51656295742487448191409429372, −11.25713092051544583063154212326, −9.349130222125409586860684392726, −6.58661284276896890433337774218, −3.59263323010505890195366133299, 4.48822831160574025841244091289, 7.07316282889632245175698041630, 8.963181990116567087645578232943, 11.61489344213984914249970358162, 13.10819895540757872456905795597, 14.50150656506435262624987352586, 16.26268387309593465847672038068, 16.98567584052670781622901093783, 19.27162279492311257663844232801, 20.23607954311365487738523774895

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