Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·3-s + 4-s − 10·5-s + 18·6-s + 14·7-s + 3·8-s + 27·9-s − 30·10-s + 62·11-s + 6·12-s − 6·13-s + 42·14-s − 60·15-s + 9·16-s + 40·17-s + 81·18-s − 122·19-s − 10·20-s + 84·21-s + 186·22-s + 16·23-s + 18·24-s + 75·25-s − 18·26-s + 108·27-s + 14·28-s + ⋯
L(s)  = 1  + 1.06·2-s + 1.15·3-s + 1/8·4-s − 0.894·5-s + 1.22·6-s + 0.755·7-s + 0.132·8-s + 9-s − 0.948·10-s + 1.69·11-s + 0.144·12-s − 0.128·13-s + 0.801·14-s − 1.03·15-s + 9/64·16-s + 0.570·17-s + 1.06·18-s − 1.47·19-s − 0.111·20-s + 0.872·21-s + 1.80·22-s + 0.145·23-s + 0.153·24-s + 3/5·25-s − 0.135·26-s + 0.769·27-s + 0.0944·28-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  =  \(0\)
Selberg data  =  \((4,\ 11025,\ (\ :3/2, 3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(5.27205\)
\(L(\frac12)\)  \(\approx\)  \(5.27205\)
\(L(\frac{5}{2})\)   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$D_{4}$ \( 1 - 3 T + p^{3} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 62 T + 3582 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 3378 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 40 T + 8750 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 122 T + 17398 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 16 T + 34 p T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 352 T + 78278 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 66 T + 45870 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 188 T + 44542 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 16 T + 18350 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 396 T + 2226 p T^{2} + 396 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 4 p T + 15582 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 982 T + 504354 T^{2} - 982 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 516 T + 418118 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 880 T + 575238 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 356 T + 100374 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 310 T + 664038 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 326 T + 789802 T^{2} - 326 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 680 T + 744870 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 796 T + 411158 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 670 T + 1145410 T^{2} + 670 p^{3} T^{3} + p^{6} T^{4} \)
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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

−13.45661982419435445974030247562, −13.42975125679078246543569448949, −12.31294012545683469886175642022, −12.25656411438848290036343603269, −11.74340904356697436312554036667, −10.99283288889158405781600786128, −10.38833133752295853137547898109, −9.849320605974813936721103228053, −8.762382598156180586820762710996, −8.760606868012696065345540490324, −8.165518684722344472648947709944, −7.41717170158120664380132296828, −6.81635547436989365399984430237, −6.24001980998692394717884292756, −4.83096030102797826311073591617, −4.71333953154907923884078271650, −3.82511888816381233528132119600, −3.57515772035303027225893561748, −2.32086255885988398607874075783, −1.20229461864017516428374702022, 1.20229461864017516428374702022, 2.32086255885988398607874075783, 3.57515772035303027225893561748, 3.82511888816381233528132119600, 4.71333953154907923884078271650, 4.83096030102797826311073591617, 6.24001980998692394717884292756, 6.81635547436989365399984430237, 7.41717170158120664380132296828, 8.165518684722344472648947709944, 8.760606868012696065345540490324, 8.762382598156180586820762710996, 9.849320605974813936721103228053, 10.38833133752295853137547898109, 10.99283288889158405781600786128, 11.74340904356697436312554036667, 12.25656411438848290036343603269, 12.31294012545683469886175642022, 13.42975125679078246543569448949, 13.45661982419435445974030247562

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