Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
Sign $0.302 - 0.953i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (18.9 + 18.9i)3-s + (−36.4 − 42.3i)5-s + (112. − 112. i)7-s + 473. i·9-s + 269. i·11-s + (−403. + 403. i)13-s + (112. − 1.49e3i)15-s + (1.09e3 + 1.09e3i)17-s + 1.80e3·19-s + 4.25e3·21-s + (2.83e3 + 2.83e3i)23-s + (−470. + 3.08e3i)25-s + (−4.35e3 + 4.35e3i)27-s − 7.85e3i·29-s + 4.68e3i·31-s + ⋯
L(s)  = 1  + (1.21 + 1.21i)3-s + (−0.651 − 0.758i)5-s + (0.867 − 0.867i)7-s + 1.94i·9-s + 0.671i·11-s + (−0.662 + 0.662i)13-s + (0.129 − 1.71i)15-s + (0.922 + 0.922i)17-s + 1.14·19-s + 2.10·21-s + (1.11 + 1.11i)23-s + (−0.150 + 0.988i)25-s + (−1.15 + 1.15i)27-s − 1.73i·29-s + 0.875i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(160\)    =    \(2^{5} \cdot 5\)
\( \varepsilon \)  =  $0.302 - 0.953i$
motivic weight  =  \(5\)
character  :  $\chi_{160} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 160,\ (\ :5/2),\ 0.302 - 0.953i)\)
\(L(3)\)  \(\approx\)  \(2.903364992\)
\(L(\frac12)\)  \(\approx\)  \(2.903364992\)
\(L(\frac{7}{2})\)   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 \( 1 + (36.4 + 42.3i)T \)
good3 \( 1 + (-18.9 - 18.9i)T + 243iT^{2} \)
7 \( 1 + (-112. + 112. i)T - 1.68e4iT^{2} \)
11 \( 1 - 269. iT - 1.61e5T^{2} \)
13 \( 1 + (403. - 403. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.09e3 - 1.09e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.80e3T + 2.47e6T^{2} \)
23 \( 1 + (-2.83e3 - 2.83e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 7.85e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.68e3iT - 2.86e7T^{2} \)
37 \( 1 + (-5.15e3 - 5.15e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.65e3T + 1.15e8T^{2} \)
43 \( 1 + (3.23e3 + 3.23e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (6.00e3 - 6.00e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (6.07e3 - 6.07e3i)T - 4.18e8iT^{2} \)
59 \( 1 + 3.93e4T + 7.14e8T^{2} \)
61 \( 1 + 1.19e4T + 8.44e8T^{2} \)
67 \( 1 + (-4.36e4 + 4.36e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 7.04e3iT - 1.80e9T^{2} \)
73 \( 1 + (-2.44e4 + 2.44e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 3.06e4T + 3.07e9T^{2} \)
83 \( 1 + (2.56e4 + 2.56e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 1.37e5iT - 5.58e9T^{2} \)
97 \( 1 + (-4.79e4 - 4.79e4i)T + 8.58e9iT^{2} \)
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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

−12.09828095692627508716758733054, −11.04266724673768714354492728551, −9.873009057012182281773591364636, −9.260784574643245707634315003230, −7.995347630275235759652980979398, −7.55539208853885057684144205480, −5.00633434107153696341632970503, −4.35895296506317126734861251303, −3.34143748452494627698279149276, −1.48185048105592344971770333785, 0.930462747598537041568561568035, 2.57115347528033955768273724667, 3.19914015631181393228383475479, 5.29399009752035148876773500012, 6.88327508343732028451115224268, 7.70920746025607811172015268174, 8.346368984724633260148750896065, 9.418297788972162149509276999358, 11.03383627892224011500169122346, 11.98086463039708733711624036397

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