Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.914 - 0.405i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.1·3-s + (−55.2 + 8.66i)5-s − 121. i·7-s + 123.·9-s + 298i·11-s + 485.·13-s + (−1.05e3 + 165. i)15-s + 420. i·17-s + 2.57e3i·19-s − 2.32e3i·21-s − 717. i·23-s + (2.97e3 − 956. i)25-s − 2.29e3·27-s − 3.73e3i·29-s + 6.22e3·31-s + ⋯
L(s)  = 1  + 1.22·3-s + (−0.987 + 0.154i)5-s − 0.937i·7-s + 0.506·9-s + 0.742i·11-s + 0.797·13-s + (−1.21 + 0.190i)15-s + 0.353i·17-s + 1.63i·19-s − 1.15i·21-s − 0.282i·23-s + (0.952 − 0.306i)25-s − 0.606·27-s − 0.824i·29-s + 1.16·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.914 - 0.405i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.914 - 0.405i)\)
\(L(3)\)  \(\approx\)  \(2.675300987\)
\(L(\frac12)\)  \(\approx\)  \(2.675300987\)
\(L(\frac{7}{2})\)   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 \( 1 + (55.2 - 8.66i)T \)
good3 \( 1 - 19.1T + 243T^{2} \)
7 \( 1 + 121. iT - 1.68e4T^{2} \)
11 \( 1 - 298iT - 1.61e5T^{2} \)
13 \( 1 - 485.T + 3.71e5T^{2} \)
17 \( 1 - 420. iT - 1.41e6T^{2} \)
19 \( 1 - 2.57e3iT - 2.47e6T^{2} \)
23 \( 1 + 717. iT - 6.43e6T^{2} \)
29 \( 1 + 3.73e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.22e3T + 2.86e7T^{2} \)
37 \( 1 - 5.72e3T + 6.93e7T^{2} \)
41 \( 1 - 1.13e4T + 1.15e8T^{2} \)
43 \( 1 - 1.91e4T + 1.47e8T^{2} \)
47 \( 1 - 2.28e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.72e4T + 4.18e8T^{2} \)
59 \( 1 + 1.63e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.87e4iT - 8.44e8T^{2} \)
67 \( 1 - 8.39e3T + 1.35e9T^{2} \)
71 \( 1 + 2.90e4T + 1.80e9T^{2} \)
73 \( 1 - 420. iT - 2.07e9T^{2} \)
79 \( 1 + 3.11e4T + 3.07e9T^{2} \)
83 \( 1 - 1.06e5T + 3.93e9T^{2} \)
89 \( 1 + 9.93e4T + 5.58e9T^{2} \)
97 \( 1 - 1.80e5iT - 8.58e9T^{2} \)
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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

−10.71870253466484715214637104937, −9.932730131739390666254592682950, −8.774119362523330123567871424819, −7.915139934287188591988055855971, −7.47030326163383992843980518251, −6.13987512981260437633852398564, −4.20791619078746892690725871352, −3.82161294533934488260216034412, −2.53690636252064932335392209528, −1.01976210223580627933231750675, 0.75438382260686888714622540711, 2.55218677988979436514775638013, 3.26190406491093730694055155232, 4.44397611558845976928808910944, 5.79476327210476489285260600290, 7.14942567497880049111525144258, 8.163542248517606020237627158002, 8.790688317941794294334149918358, 9.311639929294312268642732121848, 10.96350259839698411768371092333

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