Properties

Label 4-1575e2-1.1-c3e2-0-3
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $8635.61$
Root an. cond. $9.63991$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 7·2-s + 25·4-s + 14·7-s − 63·8-s + 26·11-s − 14·13-s − 98·14-s + 169·16-s + 16·17-s + 174·19-s − 182·22-s + 184·23-s + 98·26-s + 350·28-s + 32·29-s + 330·31-s − 623·32-s − 112·34-s + 132·37-s − 1.21e3·38-s − 200·41-s − 364·43-s + 650·44-s − 1.28e3·46-s + 292·47-s + 147·49-s − 350·52-s + ⋯
L(s)  = 1  − 2.47·2-s + 25/8·4-s + 0.755·7-s − 2.78·8-s + 0.712·11-s − 0.298·13-s − 1.87·14-s + 2.64·16-s + 0.228·17-s + 2.10·19-s − 1.76·22-s + 1.66·23-s + 0.739·26-s + 2.36·28-s + 0.204·29-s + 1.91·31-s − 3.44·32-s − 0.564·34-s + 0.586·37-s − 5.19·38-s − 0.761·41-s − 1.29·43-s + 2.22·44-s − 4.12·46-s + 0.906·47-s + 3/7·49-s − 0.933·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8635.61\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.317713835\)
\(L(\frac12)\) \(\approx\) \(1.317713835\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - p T )^{2} \)
good2$D_{4}$ \( 1 + 7 T + 3 p^{3} T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 26 T + 2406 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 14 T + 2386 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 16 T + 6558 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 174 T + 18414 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 8 p T + 19470 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 32 T + 19046 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 330 T + 85430 T^{2} - 330 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 132 T + 103214 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 200 T + 64542 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 364 T + 172486 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 292 T + 179390 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 34 T + 103410 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 364 T + 438374 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 792 T + 340886 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 788 T + 753430 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 454 T + 304526 T^{2} + 454 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 778 T + 794698 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 408 T + 994782 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1136 T + 1169990 T^{2} - 1136 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 36 T - 842170 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 498 T + 1796754 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \)
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

−9.223212462190815399640567056115, −8.942222866683829187960517739458, −8.483810764359895799836903863827, −8.213209267102001433560140016450, −7.69978051770202446239431585044, −7.67161160038063862620248009178, −6.91085803112935730018740208669, −6.85558830315182828228095917519, −6.31694576408462121717297146902, −5.57636034489855508173438237324, −5.20239369644383655317272464369, −4.93592174007107908795049562490, −4.14130297391120576408425657680, −3.55463762526887121412580851402, −2.94106904635594231010568908426, −2.64168045931014219576290578554, −1.71130330139295447440015412653, −1.28823791153560738431422382860, −0.908642947298865511065166358263, −0.53501053159481024997518667391, 0.53501053159481024997518667391, 0.908642947298865511065166358263, 1.28823791153560738431422382860, 1.71130330139295447440015412653, 2.64168045931014219576290578554, 2.94106904635594231010568908426, 3.55463762526887121412580851402, 4.14130297391120576408425657680, 4.93592174007107908795049562490, 5.20239369644383655317272464369, 5.57636034489855508173438237324, 6.31694576408462121717297146902, 6.85558830315182828228095917519, 6.91085803112935730018740208669, 7.67161160038063862620248009178, 7.69978051770202446239431585044, 8.213209267102001433560140016450, 8.483810764359895799836903863827, 8.942222866683829187960517739458, 9.223212462190815399640567056115

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