Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
Sign $0.950 + 0.310i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.28 + 9.28i)3-s + (42.6 − 36.0i)5-s + (−105. + 105. i)7-s − 70.7i·9-s − 344. i·11-s + (707. − 707. i)13-s + (731. + 61.3i)15-s + (1.26e3 + 1.26e3i)17-s + 438.·19-s − 1.96e3·21-s + (−1.72e3 − 1.72e3i)23-s + (520. − 3.08e3i)25-s + (2.91e3 − 2.91e3i)27-s − 2.51e3i·29-s + 7.14e3i·31-s + ⋯
L(s)  = 1  + (0.595 + 0.595i)3-s + (0.763 − 0.645i)5-s + (−0.816 + 0.816i)7-s − 0.291i·9-s − 0.859i·11-s + (1.16 − 1.16i)13-s + (0.839 + 0.0703i)15-s + (1.05 + 1.05i)17-s + 0.278·19-s − 0.971·21-s + (−0.679 − 0.679i)23-s + (0.166 − 0.986i)25-s + (0.768 − 0.768i)27-s − 0.554i·29-s + 1.33i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(160\)    =    \(2^{5} \cdot 5\)
\( \varepsilon \)  =  $0.950 + 0.310i$
motivic weight  =  \(5\)
character  :  $\chi_{160} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 160,\ (\ :5/2),\ 0.950 + 0.310i)\)
\(L(3)\)  \(\approx\)  \(2.658406422\)
\(L(\frac12)\)  \(\approx\)  \(2.658406422\)
\(L(\frac{7}{2})\)   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 \( 1 \)
5 \( 1 + (-42.6 + 36.0i)T \)
good3 \( 1 + (-9.28 - 9.28i)T + 243iT^{2} \)
7 \( 1 + (105. - 105. i)T - 1.68e4iT^{2} \)
11 \( 1 + 344. iT - 1.61e5T^{2} \)
13 \( 1 + (-707. + 707. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.26e3 - 1.26e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 438.T + 2.47e6T^{2} \)
23 \( 1 + (1.72e3 + 1.72e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 2.51e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.14e3iT - 2.86e7T^{2} \)
37 \( 1 + (3.36e3 + 3.36e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 6.96e3T + 1.15e8T^{2} \)
43 \( 1 + (-1.63e4 - 1.63e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.30e4 + 1.30e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.00e4 + 2.00e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 8.41e3T + 7.14e8T^{2} \)
61 \( 1 + 2.29e3T + 8.44e8T^{2} \)
67 \( 1 + (-395. + 395. i)T - 1.35e9iT^{2} \)
71 \( 1 + 4.08e4iT - 1.80e9T^{2} \)
73 \( 1 + (4.56e4 - 4.56e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 5.84e4T + 3.07e9T^{2} \)
83 \( 1 + (-2.69e4 - 2.69e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 6.19e4iT - 5.58e9T^{2} \)
97 \( 1 + (1.10e4 + 1.10e4i)T + 8.58e9iT^{2} \)
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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

−12.23899417655979869564563737581, −10.58458814689793181089726079933, −9.822805382481269672487503747421, −8.804720643531841837438126913344, −8.307664025955770678916762951489, −6.12336223343240814750730273416, −5.66809759071648031572390275729, −3.80499197063092105998737898436, −2.82907750635253006250112616136, −0.933270648023861040921065627100, 1.35857038319010044384302043991, 2.60870918179524455051715736523, 3.93428224213426673757587951389, 5.76208807610790333108515671474, 7.00255325377758710180487249546, 7.50258122259014586796350153091, 9.135623774514804789149588780481, 9.906965030754257507510853574971, 10.89962891457869383956839453365, 12.17409502133140527562389585838

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