Properties

Degree 2
Conductor 5
Sign $0.524 - 0.851i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.2 + 15.2i)2-s + (−20.3 + 20.3i)3-s + 209. i·4-s + (558. − 280. i)5-s − 620.·6-s + (−2.41e3 − 2.41e3i)7-s + (705. − 705. i)8-s + 5.73e3i·9-s + (1.28e4 + 4.24e3i)10-s − 981.·11-s + (−4.26e3 − 4.26e3i)12-s + (−2.65e4 + 2.65e4i)13-s − 7.37e4i·14-s + (−5.65e3 + 1.70e4i)15-s + 7.52e4·16-s + (−1.85e4 − 1.85e4i)17-s + ⋯
L(s)  = 1  + (0.953 + 0.953i)2-s + (−0.251 + 0.251i)3-s + 0.819i·4-s + (0.893 − 0.448i)5-s − 0.478·6-s + (−1.00 − 1.00i)7-s + (0.172 − 0.172i)8-s + 0.873i·9-s + (1.28 + 0.424i)10-s − 0.0670·11-s + (−0.205 − 0.205i)12-s + (−0.930 + 0.930i)13-s − 1.91i·14-s + (−0.111 + 0.336i)15-s + 1.14·16-s + (−0.221 − 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5\)
\( \varepsilon \)  =  $0.524 - 0.851i$
motivic weight  =  \(8\)
character  :  $\chi_{5} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 5,\ (\ :4),\ 0.524 - 0.851i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.59883 + 0.893187i\)
\(L(\frac12)\)  \(\approx\)  \(1.59883 + 0.893187i\)
\(L(5)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 5$,\(F_p(T)\) is a polynomial of degree 2. If $p = 5$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 + (-558. + 280. i)T \)
good2 \( 1 + (-15.2 - 15.2i)T + 256iT^{2} \)
3 \( 1 + (20.3 - 20.3i)T - 6.56e3iT^{2} \)
7 \( 1 + (2.41e3 + 2.41e3i)T + 5.76e6iT^{2} \)
11 \( 1 + 981.T + 2.14e8T^{2} \)
13 \( 1 + (2.65e4 - 2.65e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (1.85e4 + 1.85e4i)T + 6.97e9iT^{2} \)
19 \( 1 + 5.03e4iT - 1.69e10T^{2} \)
23 \( 1 + (-1.36e4 + 1.36e4i)T - 7.83e10iT^{2} \)
29 \( 1 - 1.05e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.09e6T + 8.52e11T^{2} \)
37 \( 1 + (-7.78e4 - 7.78e4i)T + 3.51e12iT^{2} \)
41 \( 1 - 5.54e5T + 7.98e12T^{2} \)
43 \( 1 + (1.07e6 - 1.07e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (4.06e6 + 4.06e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-1.88e6 + 1.88e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 1.27e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.40e7T + 1.91e14T^{2} \)
67 \( 1 + (9.54e6 + 9.54e6i)T + 4.06e14iT^{2} \)
71 \( 1 + 2.82e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.11e7 + 1.11e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 6.87e7iT - 1.51e15T^{2} \)
83 \( 1 + (3.29e6 - 3.29e6i)T - 2.25e15iT^{2} \)
89 \( 1 + 7.97e7iT - 3.93e15T^{2} \)
97 \( 1 + (1.96e7 + 1.96e7i)T + 7.83e15iT^{2} \)
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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

−22.62478598596543776896555696765, −21.66183205344261827386463630758, −19.60462161965874309146275334912, −16.95275000163501774460225257162, −16.20884594891809190630346450231, −14.08700637933851260423011238458, −13.07913462216372600186259149434, −10.03996946524320232548116239146, −6.80010731784266992744474803301, −4.84035359668290825174730410314, 2.76747802457835857559118822775, 5.92866625125254747864420880653, 9.927953871273442204192135892244, 12.06063191842044572364169484804, 13.14873852285603343056037849953, 14.98184253244649355601554038539, 17.53685335963925579862307146012, 19.18328799620411100995499822937, 20.90281610471816564516027301019, 22.11043543502793343001544558976

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