Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.235 + 0.971i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 28.9i·3-s + (13.1 + 54.3i)5-s + 146. i·7-s − 594.·9-s − 191.·11-s + 83.9i·13-s + (−1.57e3 + 380. i)15-s − 2.00e3i·17-s + 677.·19-s − 4.24e3·21-s − 1.29e3i·23-s + (−2.77e3 + 1.42e3i)25-s − 1.01e4i·27-s − 3.26e3·29-s + 6.15e3·31-s + ⋯
L(s)  = 1  + 1.85i·3-s + (0.235 + 0.971i)5-s + 1.13i·7-s − 2.44·9-s − 0.476·11-s + 0.137i·13-s + (−1.80 + 0.436i)15-s − 1.67i·17-s + 0.430·19-s − 2.10·21-s − 0.510i·23-s + (−0.889 + 0.457i)25-s − 2.68i·27-s − 0.721·29-s + 1.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.235 + 0.971i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.235 + 0.971i)\)
\(L(3)\)  \(\approx\)  \(0.8427631325\)
\(L(\frac12)\)  \(\approx\)  \(0.8427631325\)
\(L(\frac{7}{2})\)   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 \( 1 + (-13.1 - 54.3i)T \)
good3 \( 1 - 28.9iT - 243T^{2} \)
7 \( 1 - 146. iT - 1.68e4T^{2} \)
11 \( 1 + 191.T + 1.61e5T^{2} \)
13 \( 1 - 83.9iT - 3.71e5T^{2} \)
17 \( 1 + 2.00e3iT - 1.41e6T^{2} \)
19 \( 1 - 677.T + 2.47e6T^{2} \)
23 \( 1 + 1.29e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.26e3T + 2.05e7T^{2} \)
31 \( 1 - 6.15e3T + 2.86e7T^{2} \)
37 \( 1 - 1.13e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.05e4T + 1.15e8T^{2} \)
43 \( 1 - 1.29e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.52e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.47e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.82e4T + 7.14e8T^{2} \)
61 \( 1 - 3.58e3T + 8.44e8T^{2} \)
67 \( 1 + 2.17e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.13e4T + 1.80e9T^{2} \)
73 \( 1 - 1.33e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.59e4T + 3.07e9T^{2} \)
83 \( 1 + 5.33e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.13e4T + 5.58e9T^{2} \)
97 \( 1 - 8.08e4iT - 8.58e9T^{2} \)
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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

−11.36888658169014918641714290056, −10.47509576234030495660559697822, −9.670471577851944699601101636517, −9.146216083190282372851735611590, −7.966138744420814012754638498487, −6.41920076102096641188422721543, −5.39975285190992694012480437009, −4.64622546463512280787081419634, −3.14794869640917152669270837185, −2.63699588327943709327990799062, 0.22833902311978010862296414529, 1.17828461587320731254994964167, 2.03968670472860891852103169549, 3.73676598956550360194402668012, 5.32840478218395409028792085268, 6.24026730283820534866658788907, 7.29398209034771539277155951564, 7.985140430034919595764272689683, 8.729073336372853418796554218307, 10.13178454727591890434916090370

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