Properties

Degree 4
Conductor $ 3^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 7
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·2-s + 54·3-s − 69·4-s + 250·5-s + 378·6-s + 1.30e3·7-s − 413·8-s + 2.18e3·9-s + 1.75e3·10-s + 3.44e3·11-s − 3.72e3·12-s − 8.98e3·13-s + 9.12e3·14-s + 1.35e4·15-s − 4.86e3·16-s − 5.49e3·17-s + 1.53e4·18-s − 4.95e4·19-s − 1.72e4·20-s + 7.04e4·21-s + 2.41e4·22-s + 9.18e4·23-s − 2.23e4·24-s + 4.68e4·25-s − 6.29e4·26-s + 7.87e4·27-s − 8.99e4·28-s + ⋯
L(s)  = 1  + 0.618·2-s + 1.15·3-s − 0.539·4-s + 0.894·5-s + 0.714·6-s + 1.43·7-s − 0.285·8-s + 9-s + 0.553·10-s + 0.781·11-s − 0.622·12-s − 1.13·13-s + 0.889·14-s + 1.03·15-s − 0.296·16-s − 0.271·17-s + 0.618·18-s − 1.65·19-s − 0.482·20-s + 1.65·21-s + 0.483·22-s + 1.57·23-s − 0.329·24-s + 3/5·25-s − 0.702·26-s + 0.769·27-s − 0.774·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(225\)    =    \(3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  induced by $\chi_{15} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 225,\ (\ :7/2, 7/2),\ 1)$
$L(4)$  $\approx$  $4.47880$
$L(\frac12)$  $\approx$  $4.47880$
$L(\frac{9}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{3,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( ( 1 - p^{3} T )^{2} \)
5$C_1$ \( ( 1 - p^{3} T )^{2} \)
good2$D_{4}$ \( 1 - 7 T + 59 p T^{2} - 7 p^{7} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 - 1304 T + 1601006 T^{2} - 1304 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 - 3448 T + 9598294 T^{2} - 3448 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 8988 T + 43676926 T^{2} + 8988 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 + 5492 T + 533727862 T^{2} + 5492 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 + 49584 T + 1767259558 T^{2} + 49584 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 - 91848 T + 4843394254 T^{2} - 91848 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 - 6268 p T + 32439203278 T^{2} - 6268 p^{8} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 304232 T + 77458297022 T^{2} - 304232 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 502316 T + 221684315886 T^{2} + 502316 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 631172 T + 420346017142 T^{2} - 631172 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 353640 T + 567251429590 T^{2} - 353640 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 + 467480 T + 1062629128990 T^{2} + 467480 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 + 568052 T + 2403786403294 T^{2} + 568052 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 287224 T + 1627391637238 T^{2} - 287224 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 2514180 T + 7865442419758 T^{2} + 2514180 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 + 5073832 T + 16021265240102 T^{2} + 5073832 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 + 3748816 T + 20824804809646 T^{2} + 3748816 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 1477212 T - 3158190986 T^{2} + 1477212 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 + 4627720 T + 42789559383518 T^{2} + 4627720 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 + 6072936 T + 62224060120582 T^{2} + 6072936 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 16516356 T + 156597055746838 T^{2} - 16516356 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 2723428 T + 135845063471622 T^{2} - 2723428 p^{7} T^{3} + p^{14} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.74919663695131842440082232171, −17.55842006008268944748005967664, −17.04877873582142923487452575919, −15.79105659830712106912656777392, −14.78061773736410346579605067880, −14.62274834405699445967999911824, −14.07335447537991267461008271188, −13.47892113249999878498918010377, −12.80851836828137777524822912202, −11.99031239457933918155157372866, −10.83335945020528956171161022592, −10.06152442372522655314365043787, −8.895535089709248355697539610103, −8.760884674874884043127220695129, −7.54029388453871757024335302251, −6.44363249489805156692362086648, −4.66131089137715335268124592628, −4.62351224268796633110936965482, −2.70596641029739944237853956738, −1.55323263000999941611920540855, 1.55323263000999941611920540855, 2.70596641029739944237853956738, 4.62351224268796633110936965482, 4.66131089137715335268124592628, 6.44363249489805156692362086648, 7.54029388453871757024335302251, 8.760884674874884043127220695129, 8.895535089709248355697539610103, 10.06152442372522655314365043787, 10.83335945020528956171161022592, 11.99031239457933918155157372866, 12.80851836828137777524822912202, 13.47892113249999878498918010377, 14.07335447537991267461008271188, 14.62274834405699445967999911824, 14.78061773736410346579605067880, 15.79105659830712106912656777392, 17.04877873582142923487452575919, 17.55842006008268944748005967664, 17.74919663695131842440082232171

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