Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 5 $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25·5-s − 62·7-s + 144·11-s − 654·13-s + 1.19e3·17-s + 556·19-s − 2.18e3·23-s + 625·25-s + 1.57e3·29-s + 9.66e3·31-s − 1.55e3·35-s − 3.53e3·37-s − 7.46e3·41-s − 7.11e3·43-s + 2.82e4·47-s − 1.29e4·49-s + 1.30e4·53-s + 3.60e3·55-s + 3.70e4·59-s + 3.95e4·61-s − 1.63e4·65-s − 5.67e4·67-s − 4.55e4·71-s + 1.18e4·73-s − 8.92e3·77-s + 9.42e4·79-s + 3.14e4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.478·7-s + 0.358·11-s − 1.07·13-s + 0.998·17-s + 0.353·19-s − 0.860·23-s + 1/5·25-s + 0.348·29-s + 1.80·31-s − 0.213·35-s − 0.424·37-s − 0.693·41-s − 0.586·43-s + 1.86·47-s − 0.771·49-s + 0.637·53-s + 0.160·55-s + 1.38·59-s + 1.36·61-s − 0.479·65-s − 1.54·67-s − 1.07·71-s + 0.260·73-s − 0.171·77-s + 1.69·79-s + 0.501·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{360} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 360,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(2.074348642\)
\(L(\frac12)\)  \(\approx\)  \(2.074348642\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - p^{2} T \)
good7 \( 1 + 62 T + p^{5} T^{2} \)
11 \( 1 - 144 T + p^{5} T^{2} \)
13 \( 1 + 654 T + p^{5} T^{2} \)
17 \( 1 - 70 p T + p^{5} T^{2} \)
19 \( 1 - 556 T + p^{5} T^{2} \)
23 \( 1 + 2182 T + p^{5} T^{2} \)
29 \( 1 - 1578 T + p^{5} T^{2} \)
31 \( 1 - 9660 T + p^{5} T^{2} \)
37 \( 1 + 3534 T + p^{5} T^{2} \)
41 \( 1 + 182 p T + p^{5} T^{2} \)
43 \( 1 + 7114 T + p^{5} T^{2} \)
47 \( 1 - 602 p T + p^{5} T^{2} \)
53 \( 1 - 13046 T + p^{5} T^{2} \)
59 \( 1 - 37092 T + p^{5} T^{2} \)
61 \( 1 - 39570 T + p^{5} T^{2} \)
67 \( 1 + 56734 T + p^{5} T^{2} \)
71 \( 1 + 45588 T + p^{5} T^{2} \)
73 \( 1 - 11842 T + p^{5} T^{2} \)
79 \( 1 - 94216 T + p^{5} T^{2} \)
83 \( 1 - 31482 T + p^{5} T^{2} \)
89 \( 1 - 94054 T + p^{5} T^{2} \)
97 \( 1 - 23714 T + p^{5} T^{2} \)
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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

−10.19131959147977562580842439262, −9.971857605728331527751048099259, −8.848553775367841748859549177250, −7.75690650264061333366615703474, −6.76385430739572280141648619794, −5.79989847540953640044860870936, −4.74460007654650185417604275880, −3.40639657220047482603143127182, −2.24856837449652082567304116334, −0.78044757537213289225227121995, 0.78044757537213289225227121995, 2.24856837449652082567304116334, 3.40639657220047482603143127182, 4.74460007654650185417604275880, 5.79989847540953640044860870936, 6.76385430739572280141648619794, 7.75690650264061333366615703474, 8.848553775367841748859549177250, 9.971857605728331527751048099259, 10.19131959147977562580842439262

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