Properties

Degree $4$
Conductor $275625$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 6·3-s + 4-s + 18·6-s − 14·7-s − 3·8-s + 27·9-s + 62·11-s − 6·12-s + 6·13-s + 42·14-s + 9·16-s − 40·17-s − 81·18-s − 122·19-s + 84·21-s − 186·22-s − 16·23-s + 18·24-s − 18·26-s − 108·27-s − 14·28-s + 352·29-s + 66·31-s + 165·32-s − 372·33-s + 120·34-s + ⋯
L(s)  = 1  − 1.06·2-s − 1.15·3-s + 1/8·4-s + 1.22·6-s − 0.755·7-s − 0.132·8-s + 9-s + 1.69·11-s − 0.144·12-s + 0.128·13-s + 0.801·14-s + 9/64·16-s − 0.570·17-s − 1.06·18-s − 1.47·19-s + 0.872·21-s − 1.80·22-s − 0.145·23-s + 0.153·24-s − 0.135·26-s − 0.769·27-s − 0.0944·28-s + 2.25·29-s + 0.382·31-s + 0.911·32-s − 1.96·33-s + 0.605·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(275625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 275625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8435282514\)
\(L(\frac12)\) \(\approx\) \(0.8435282514\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + p T )^{2} \)
5 \( 1 \)
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$D_{4}$ \( 1 - 670 T + 1145410 T^{2} - 670 p^{3} T^{3} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.44952857988875015793601439060, −10.40099230716399718473465553066, −9.540365068206106608860730546166, −9.514644768178388124076130362418, −8.873951856178545941469969180437, −8.778035039137146358886626658437, −7.998355039869237451727281040568, −7.66450475351796549017430363913, −6.72590460896638801413599308830, −6.46481252689732640356209446685, −6.34581754091404579682857301941, −5.94609151761960009021876136681, −4.91997094614412997784028654818, −4.47848582870770620047149031389, −4.13752717693359713465153361429, −3.36926902953674733975486515684, −2.55337736460355621813102975690, −1.73711471114374038831705788498, −0.66305326649196143557993196086, −0.66253792770971114502580459844, 0.66253792770971114502580459844, 0.66305326649196143557993196086, 1.73711471114374038831705788498, 2.55337736460355621813102975690, 3.36926902953674733975486515684, 4.13752717693359713465153361429, 4.47848582870770620047149031389, 4.91997094614412997784028654818, 5.94609151761960009021876136681, 6.34581754091404579682857301941, 6.46481252689732640356209446685, 6.72590460896638801413599308830, 7.66450475351796549017430363913, 7.998355039869237451727281040568, 8.778035039137146358886626658437, 8.873951856178545941469969180437, 9.514644768178388124076130362418, 9.540365068206106608860730546166, 10.40099230716399718473465553066, 10.44952857988875015793601439060

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