Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $0.987 - 0.156i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28.5·3-s + (5.81 − 55.5i)5-s − 92.3i·7-s + 571.·9-s + 541. i·11-s − 782.·13-s + (−165. + 1.58e3i)15-s − 1.55e3i·17-s + 484. i·19-s + 2.63e3i·21-s + 1.09e3i·23-s + (−3.05e3 − 646. i)25-s − 9.37e3·27-s + 597. i·29-s − 8.86e3·31-s + ⋯
L(s)  = 1  − 1.83·3-s + (0.104 − 0.994i)5-s − 0.712i·7-s + 2.35·9-s + 1.34i·11-s − 1.28·13-s + (−0.190 + 1.82i)15-s − 1.30i·17-s + 0.307i·19-s + 1.30i·21-s + 0.432i·23-s + (−0.978 − 0.206i)25-s − 2.47·27-s + 0.131i·29-s − 1.65·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $0.987 - 0.156i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ 0.987 - 0.156i)\)
\(L(3)\)  \(\approx\)  \(0.5522652773\)
\(L(\frac12)\)  \(\approx\)  \(0.5522652773\)
\(L(\frac{7}{2})\)   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 \( 1 + (-5.81 + 55.5i)T \)
good3 \( 1 + 28.5T + 243T^{2} \)
7 \( 1 + 92.3iT - 1.68e4T^{2} \)
11 \( 1 - 541. iT - 1.61e5T^{2} \)
13 \( 1 + 782.T + 3.71e5T^{2} \)
17 \( 1 + 1.55e3iT - 1.41e6T^{2} \)
19 \( 1 - 484. iT - 2.47e6T^{2} \)
23 \( 1 - 1.09e3iT - 6.43e6T^{2} \)
29 \( 1 - 597. iT - 2.05e7T^{2} \)
31 \( 1 + 8.86e3T + 2.86e7T^{2} \)
37 \( 1 + 1.36e4T + 6.93e7T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 + 1.82e3T + 1.47e8T^{2} \)
47 \( 1 - 1.79e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.65e3T + 4.18e8T^{2} \)
59 \( 1 + 1.44e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.45e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.68e4T + 1.35e9T^{2} \)
71 \( 1 + 4.80e4T + 1.80e9T^{2} \)
73 \( 1 - 6.06e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.81e4T + 3.07e9T^{2} \)
83 \( 1 - 2.45e4T + 3.93e9T^{2} \)
89 \( 1 - 1.17e5T + 5.58e9T^{2} \)
97 \( 1 - 3.77e4iT - 8.58e9T^{2} \)
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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

−10.89207372750616113123928398490, −9.937254957157555585156279299950, −9.394996014448973586727677332256, −7.39816113102960967591825365393, −7.11528366329183409407683993968, −5.60835751038447259600255157580, −4.93631951511687021297091912476, −4.25248327414960150274676876526, −1.77888798406202610632934701803, −0.58826763222732401854033349528, 0.33936761427805172310649067783, 2.06892080171511769766836051235, 3.66982535080899838350088346941, 5.16441622498499821581068897004, 5.87532291819778543270074133210, 6.58709294403100546014586470409, 7.57660886846478160965084453628, 9.070797612126215510609677831575, 10.38910195181500576580461006182, 10.73375564869658595346636787381

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