Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 6·3-s − 5·4-s + 10·5-s + 12·6-s − 14·7-s + 12·8-s + 27·9-s − 20·10-s − 16·11-s + 30·12-s − 76·13-s + 28·14-s − 60·15-s − 11·16-s − 124·17-s − 54·18-s − 96·19-s − 50·20-s + 84·21-s + 32·22-s − 16·23-s − 72·24-s + 75·25-s + 152·26-s − 108·27-s + 70·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 5/8·4-s + 0.894·5-s + 0.816·6-s − 0.755·7-s + 0.530·8-s + 9-s − 0.632·10-s − 0.438·11-s + 0.721·12-s − 1.62·13-s + 0.534·14-s − 1.03·15-s − 0.171·16-s − 1.76·17-s − 0.707·18-s − 1.15·19-s − 0.559·20-s + 0.872·21-s + 0.310·22-s − 0.145·23-s − 0.612·24-s + 3/5·25-s + 1.14·26-s − 0.769·27-s + 0.472·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{105} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 11025,\ (\ :3/2, 3/2),\ 1)\)
\(L(2)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + p T )^{2} \)
5$C_1$ \( ( 1 - p T )^{2} \)
7$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 + p T + 9 T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 16 T - 474 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 76 T + 5806 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 124 T + 13638 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 96 T + 13974 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 16 T + 15150 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 188 T + 34286 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 120 T + 60590 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 132 T + 70814 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 100 T + 89142 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 536 T + 200086 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 928 T + 408830 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 884 T + 460350 T^{2} - 884 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 104 T + 80534 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 468 T + 494606 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1688 T + 1302310 T^{2} + 1688 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 136 T + 540446 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 508 T + 13078 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 432 T + 602142 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 584 T + 1172390 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1404 T + 1802390 T^{2} + 1404 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1188 T + 2161254 T^{2} + 1188 p^{3} T^{3} + p^{6} T^{4} \)
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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

−13.08292458916580721099457241307, −12.52362329578303523174612218328, −11.98572694454539870279324252193, −11.40906351113754730786648519606, −10.58069027880479118600035779748, −10.35731378291167851175667760938, −9.718913732199968094925560978861, −9.522381603063368539181413992794, −8.655846868553477110650215124948, −8.382732867907255423730636167999, −7.15431105242583476581047669936, −6.78724653057891845595441544403, −6.25238828112403108123191162025, −5.46566543781079009701801300000, −4.70989698121031038534607877438, −4.41405209067270110225811331286, −2.82999919904791572750541919205, −1.85312014652769458321998380111, 0, 0, 1.85312014652769458321998380111, 2.82999919904791572750541919205, 4.41405209067270110225811331286, 4.70989698121031038534607877438, 5.46566543781079009701801300000, 6.25238828112403108123191162025, 6.78724653057891845595441544403, 7.15431105242583476581047669936, 8.382732867907255423730636167999, 8.655846868553477110650215124948, 9.522381603063368539181413992794, 9.718913732199968094925560978861, 10.35731378291167851175667760938, 10.58069027880479118600035779748, 11.40906351113754730786648519606, 11.98572694454539870279324252193, 12.52362329578303523174612218328, 13.08292458916580721099457241307

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