Properties

Degree 2
Conductor $ 2 \cdot 5 $
Sign $1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 174·3-s + 256·4-s − 625·5-s + 2.78e3·6-s + 4.65e3·7-s + 4.09e3·8-s + 1.05e4·9-s − 1.00e4·10-s + 2.89e4·11-s + 4.45e4·12-s − 1.64e5·13-s + 7.45e4·14-s − 1.08e5·15-s + 6.55e4·16-s − 5.94e5·17-s + 1.69e5·18-s − 2.95e5·19-s − 1.60e5·20-s + 8.10e5·21-s + 4.63e5·22-s + 2.54e6·23-s + 7.12e5·24-s + 3.90e5·25-s − 2.63e6·26-s − 1.58e6·27-s + 1.19e6·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.24·3-s + 1/2·4-s − 0.447·5-s + 0.876·6-s + 0.733·7-s + 0.353·8-s + 0.538·9-s − 0.316·10-s + 0.597·11-s + 0.620·12-s − 1.59·13-s + 0.518·14-s − 0.554·15-s + 1/4·16-s − 1.72·17-s + 0.380·18-s − 0.520·19-s − 0.223·20-s + 0.909·21-s + 0.422·22-s + 1.89·23-s + 0.438·24-s + 1/5·25-s − 1.12·26-s − 0.572·27-s + 0.366·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(9\)
character  :  $\chi_{10} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 10,\ (\ :9/2),\ 1)$
$L(5)$  $\approx$  $2.99287$
$L(\frac12)$  $\approx$  $2.99287$
$L(\frac{11}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - p^{4} T \)
5 \( 1 + p^{4} T \)
good3 \( 1 - 58 p T + p^{9} T^{2} \)
7 \( 1 - 4658 T + p^{9} T^{2} \)
11 \( 1 - 28992 T + p^{9} T^{2} \)
13 \( 1 + 164446 T + p^{9} T^{2} \)
17 \( 1 + 594822 T + p^{9} T^{2} \)
19 \( 1 + 295780 T + p^{9} T^{2} \)
23 \( 1 - 2544534 T + p^{9} T^{2} \)
29 \( 1 + 3722970 T + p^{9} T^{2} \)
31 \( 1 - 2335772 T + p^{9} T^{2} \)
37 \( 1 - 10840418 T + p^{9} T^{2} \)
41 \( 1 - 21593862 T + p^{9} T^{2} \)
43 \( 1 - 10832294 T + p^{9} T^{2} \)
47 \( 1 - 5172138 T + p^{9} T^{2} \)
53 \( 1 - 98179674 T + p^{9} T^{2} \)
59 \( 1 - 16162860 T + p^{9} T^{2} \)
61 \( 1 + 43928158 T + p^{9} T^{2} \)
67 \( 1 + 81557422 T + p^{9} T^{2} \)
71 \( 1 - 161307732 T + p^{9} T^{2} \)
73 \( 1 + 247147966 T + p^{9} T^{2} \)
79 \( 1 + 583345720 T + p^{9} T^{2} \)
83 \( 1 + 14571786 T + p^{9} T^{2} \)
89 \( 1 - 470133690 T + p^{9} T^{2} \)
97 \( 1 + 117838462 T + p^{9} T^{2} \)
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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

−19.33610689019409058756672407773, −17.18783510833343306664385553500, −15.12536578225076697366369220001, −14.56928737797231719151061539377, −13.05526482336823790486228763245, −11.32613573835874763154740711097, −8.974640954207995803283521828890, −7.33098920250425871494587907064, −4.44860326207372219395131366234, −2.46520826210613341912466793341, 2.46520826210613341912466793341, 4.44860326207372219395131366234, 7.33098920250425871494587907064, 8.974640954207995803283521828890, 11.32613573835874763154740711097, 13.05526482336823790486228763245, 14.56928737797231719151061539377, 15.12536578225076697366369220001, 17.18783510833343306664385553500, 19.33610689019409058756672407773

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