Properties

Degree 2
Conductor $ 2 \cdot 5 $
Sign $0.894 - 0.447i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 2i·3-s − 4·4-s + (−5 − 10i)5-s + 4·6-s + 26i·7-s − 8i·8-s + 23·9-s + (20 − 10i)10-s − 28·11-s + 8i·12-s − 12i·13-s − 52·14-s + (−20 + 10i)15-s + 16·16-s − 64i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.384i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s + 0.272·6-s + 1.40i·7-s − 0.353i·8-s + 0.851·9-s + (0.632 − 0.316i)10-s − 0.767·11-s + 0.192i·12-s − 0.256i·13-s − 0.992·14-s + (−0.344 + 0.172i)15-s + 0.250·16-s − 0.913i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $0.894 - 0.447i$
motivic weight  =  \(3\)
character  :  $\chi_{10} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 10,\ (\ :3/2),\ 0.894 - 0.447i)$
$L(2)$  $\approx$  $0.793233 + 0.187257i$
$L(\frac12)$  $\approx$  $0.793233 + 0.187257i$
$L(\frac{5}{2})$   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 \( 1 - 2iT \)
5 \( 1 + (5 + 10i)T \)
good3 \( 1 + 2iT - 27T^{2} \)
7 \( 1 - 26iT - 343T^{2} \)
11 \( 1 + 28T + 1.33e3T^{2} \)
13 \( 1 + 12iT - 2.19e3T^{2} \)
17 \( 1 + 64iT - 4.91e3T^{2} \)
19 \( 1 - 60T + 6.85e3T^{2} \)
23 \( 1 - 58iT - 1.21e4T^{2} \)
29 \( 1 + 90T + 2.43e4T^{2} \)
31 \( 1 + 128T + 2.97e4T^{2} \)
37 \( 1 - 236iT - 5.06e4T^{2} \)
41 \( 1 - 242T + 6.89e4T^{2} \)
43 \( 1 + 362iT - 7.95e4T^{2} \)
47 \( 1 - 226iT - 1.03e5T^{2} \)
53 \( 1 - 108iT - 1.48e5T^{2} \)
59 \( 1 - 20T + 2.05e5T^{2} \)
61 \( 1 - 542T + 2.26e5T^{2} \)
67 \( 1 + 434iT - 3.00e5T^{2} \)
71 \( 1 + 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + 632iT - 3.89e5T^{2} \)
79 \( 1 - 720T + 4.93e5T^{2} \)
83 \( 1 - 478iT - 5.71e5T^{2} \)
89 \( 1 - 490T + 7.04e5T^{2} \)
97 \( 1 - 1.45e3iT - 9.12e5T^{2} \)
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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

−20.75404921372514853975775707566, −18.93994862899296652043141094959, −18.00134541967902288543044622023, −16.12713599752384189961533448556, −15.36501571992824630772995142126, −13.24272907549937557694067291051, −12.08341892017131722999543459693, −9.264129044356701969990396274366, −7.68917674492804863897978674182, −5.30666260524625395791180327754, 3.96135615060000715278264354271, 7.43215939908228079028033812646, 10.10423520548203916488377750395, 10.98819281810985439264008946563, 13.04587947367273779802550776539, 14.56566335222668874528233179028, 16.21971445973795486680043857101, 17.98633281071982953329448072587, 19.24234170159821737585396352626, 20.49393296500168893853370391538

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