Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.81i·3-s − 55.9·5-s − 171. i·7-s + 181.·9-s − 437. i·15-s + 1.34e3·21-s + 230. i·23-s + 3.12e3·25-s + 3.32e3i·27-s − 1.68e3·29-s + 9.60e3i·35-s − 2.10e4·41-s + 2.20e4i·43-s − 1.01e4·45-s − 3.01e4i·47-s + ⋯
L(s)  = 1  + 0.501i·3-s − 0.999·5-s − 1.32i·7-s + 0.748·9-s − 0.501i·15-s + 0.664·21-s + 0.0909i·23-s + 25-s + 0.877i·27-s − 0.372·29-s + 1.32i·35-s − 1.95·41-s + 1.81i·43-s − 0.748·45-s − 1.98i·47-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(\approx\)  \(0.05341236993\)
\(L(\frac12)\)  \(\approx\)  \(0.05341236993\)
\(L(\frac{7}{2})\)   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 \( 1 + 55.9T \)
good3 \( 1 - 7.81iT - 243T^{2} \)
7 \( 1 + 171. iT - 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 - 1.41e6T^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 - 230. iT - 6.43e6T^{2} \)
29 \( 1 + 1.68e3T + 2.05e7T^{2} \)
31 \( 1 + 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 + 2.10e4T + 1.15e8T^{2} \)
43 \( 1 - 2.20e4iT - 1.47e8T^{2} \)
47 \( 1 + 3.01e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 + 5.22e4T + 8.44e8T^{2} \)
67 \( 1 + 3.68e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 + 3.07e9T^{2} \)
83 \( 1 - 4.67e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.49e5T + 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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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

−10.41641828193682378388240073062, −9.582407280175819048561075445926, −8.310380374916659255994447365199, −7.41497957786943710077385622048, −6.71800909600710476091120120081, −4.95357823216529799456999464621, −4.12800083073918679373554680814, −3.37217094144573481599812152901, −1.31924340073580256312494835348, −0.01491690321055394489243904966, 1.56051818962902983714344564407, 2.86892059711192954432102364881, 4.17489236523629184433653033193, 5.34749580260020109219258544268, 6.55386298005471834313459555987, 7.47970700466474966682614620559, 8.373139326743262054517025353630, 9.207007561989654662546359031829, 10.41084693131983839749203892519, 11.52135975722655846829996227888

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