Properties

Degree 2
Conductor $ 2 \cdot 5 $
Sign $-0.598 - 0.801i$
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 + (−39.7 + 39.7i)3-s + 128i·4-s + (−401. + 479. i)5-s − 636.·6-s + (144. + 144. i)7-s + (−1.02e3 + 1.02e3i)8-s + 3.39e3i·9-s + (−7.04e3 + 621. i)10-s + 1.36e4·11-s + (−5.09e3 − 5.09e3i)12-s + (3.08e4 − 3.08e4i)13-s + 2.31e3i·14-s + (−3.09e3 − 3.50e4i)15-s − 1.63e4·16-s + (4.94e3 + 4.94e3i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.491 + 0.491i)3-s + 0.5i·4-s + (−0.642 + 0.766i)5-s − 0.491·6-s + (0.0601 + 0.0601i)7-s + (−0.250 + 0.250i)8-s + 0.517i·9-s + (−0.704 + 0.0621i)10-s + 0.928·11-s + (−0.245 − 0.245i)12-s + (1.08 − 1.08i)13-s + 0.0601i·14-s + (−0.0610 − 0.691i)15-s − 0.250·16-s + (0.0592 + 0.0592i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(9-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $-0.598 - 0.801i$
motivic weight  =  \(8\)
character  :  $\chi_{10} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 10,\ (\ :4),\ -0.598 - 0.801i)$
$L(\frac{9}{2})$  $\approx$  $0.637158 + 1.27134i$
$L(\frac12)$  $\approx$  $0.637158 + 1.27134i$
$L(5)$   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 \( 1 + (401. - 479. i)T \)
good3 \( 1 + (39.7 - 39.7i)T - 6.56e3iT^{2} \)
7 \( 1 + (-144. - 144. i)T + 5.76e6iT^{2} \)
11 \( 1 - 1.36e4T + 2.14e8T^{2} \)
13 \( 1 + (-3.08e4 + 3.08e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-4.94e3 - 4.94e3i)T + 6.97e9iT^{2} \)
19 \( 1 - 1.76e5iT - 1.69e10T^{2} \)
23 \( 1 + (2.29e5 - 2.29e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 1.44e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.50e6T + 8.52e11T^{2} \)
37 \( 1 + (1.60e6 + 1.60e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 3.62e6T + 7.98e12T^{2} \)
43 \( 1 + (5.75e5 - 5.75e5i)T - 1.16e13iT^{2} \)
47 \( 1 + (-3.11e6 - 3.11e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-8.16e6 + 8.16e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 1.54e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.92e7T + 1.91e14T^{2} \)
67 \( 1 + (1.00e7 + 1.00e7i)T + 4.06e14iT^{2} \)
71 \( 1 - 1.84e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.45e7 + 2.45e7i)T - 8.06e14iT^{2} \)
79 \( 1 + 2.17e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.00e6 + 4.00e6i)T - 2.25e15iT^{2} \)
89 \( 1 - 6.14e7iT - 3.93e15T^{2} \)
97 \( 1 + (-1.55e7 - 1.55e7i)T + 7.83e15iT^{2} \)
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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

−19.51173591039567534973928063405, −17.88928497174011463984732795914, −16.35765836656535190721152744493, −15.34134078252824709278946876534, −13.92637399069533097122386065109, −11.92433843335304081474619900785, −10.53718074511801605687081798906, −7.946550086625866825545912688134, −5.96129765775866742918006259161, −3.82020611421012658503321505391, 0.986585465502458767950738782641, 4.21444491938491545766826703369, 6.48503423403572714390838654199, 8.986740064601155599554704885513, 11.42045660839704419276426396280, 12.25550210275362855451451000886, 13.78214722041872119100607104873, 15.62055112811890342928267547497, 17.15867894954235647710490256542, 18.74899178375951957395398157140

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