Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 24·3-s + 16·4-s + 25·5-s − 96·6-s − 172·7-s − 64·8-s + 333·9-s − 100·10-s + 132·11-s + 384·12-s − 946·13-s + 688·14-s + 600·15-s + 256·16-s − 222·17-s − 1.33e3·18-s + 500·19-s + 400·20-s − 4.12e3·21-s − 528·22-s + 3.56e3·23-s − 1.53e3·24-s + 625·25-s + 3.78e3·26-s + 2.16e3·27-s − 2.75e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.53·3-s + 1/2·4-s + 0.447·5-s − 1.08·6-s − 1.32·7-s − 0.353·8-s + 1.37·9-s − 0.316·10-s + 0.328·11-s + 0.769·12-s − 1.55·13-s + 0.938·14-s + 0.688·15-s + 1/4·16-s − 0.186·17-s − 0.968·18-s + 0.317·19-s + 0.223·20-s − 2.04·21-s − 0.232·22-s + 1.40·23-s − 0.544·24-s + 1/5·25-s + 1.09·26-s + 0.570·27-s − 0.663·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{10} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 10,\ (\ :5/2),\ 1)$
$L(3)$  $\approx$  $1.23952$
$L(\frac12)$  $\approx$  $1.23952$
$L(\frac{7}{2})$   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 \( 1 + p^{2} T \)
5 \( 1 - p^{2} T \)
good3 \( 1 - 8 p T + p^{5} T^{2} \)
7 \( 1 + 172 T + p^{5} T^{2} \)
11 \( 1 - 12 p T + p^{5} T^{2} \)
13 \( 1 + 946 T + p^{5} T^{2} \)
17 \( 1 + 222 T + p^{5} T^{2} \)
19 \( 1 - 500 T + p^{5} T^{2} \)
23 \( 1 - 3564 T + p^{5} T^{2} \)
29 \( 1 - 2190 T + p^{5} T^{2} \)
31 \( 1 - 2312 T + p^{5} T^{2} \)
37 \( 1 + 11242 T + p^{5} T^{2} \)
41 \( 1 - 1242 T + p^{5} T^{2} \)
43 \( 1 - 20624 T + p^{5} T^{2} \)
47 \( 1 - 6588 T + p^{5} T^{2} \)
53 \( 1 + 21066 T + p^{5} T^{2} \)
59 \( 1 - 7980 T + p^{5} T^{2} \)
61 \( 1 - 16622 T + p^{5} T^{2} \)
67 \( 1 - 1808 T + p^{5} T^{2} \)
71 \( 1 + 24528 T + p^{5} T^{2} \)
73 \( 1 - 20474 T + p^{5} T^{2} \)
79 \( 1 + 46240 T + p^{5} T^{2} \)
83 \( 1 + 51576 T + p^{5} T^{2} \)
89 \( 1 + 110310 T + p^{5} T^{2} \)
97 \( 1 + 78382 T + p^{5} T^{2} \)
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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

−19.59391109551468014927886506832, −19.12123466791784322961257894419, −17.18326538768877506878988180891, −15.58982682437729295459342374829, −14.22814262916776208556752957778, −12.71552430934689029751480805956, −9.914723701378331839439863103411, −9.025882871241575093930837269361, −7.12322107097916313080871502614, −2.80721942397989604211213493727, 2.80721942397989604211213493727, 7.12322107097916313080871502614, 9.025882871241575093930837269361, 9.914723701378331839439863103411, 12.71552430934689029751480805956, 14.22814262916776208556752957778, 15.58982682437729295459342374829, 17.18326538768877506878988180891, 19.12123466791784322961257894419, 19.59391109551468014927886506832

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