Properties

Label 4-1575e2-1.1-c3e2-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 4-s + 14·7-s + 24·8-s + 92·11-s − 8·13-s − 56·14-s − 47·16-s − 44·17-s − 108·19-s − 368·22-s − 320·23-s + 32·26-s + 14·28-s + 236·29-s − 60·31-s + 52·32-s + 176·34-s − 204·37-s + 432·38-s − 44·41-s − 136·43-s + 92·44-s + 1.28e3·46-s + 400·47-s + 147·49-s − 8·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/8·4-s + 0.755·7-s + 1.06·8-s + 2.52·11-s − 0.170·13-s − 1.06·14-s − 0.734·16-s − 0.627·17-s − 1.30·19-s − 3.56·22-s − 2.90·23-s + 0.241·26-s + 0.0944·28-s + 1.51·29-s − 0.347·31-s + 0.287·32-s + 0.887·34-s − 0.906·37-s + 1.84·38-s − 0.167·41-s − 0.482·43-s + 0.315·44-s + 4.10·46-s + 1.24·47-s + 3/7·49-s − 0.0213·52-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

\(L(2)\) \(\approx\) \(1.050428496\)
\(L(\frac12)\) \(\approx\) \(1.050428496\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 + p^{2} T + 15 T^{2} + p^{5} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 92 T + 4758 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 8 T - 2810 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 44 T + 630 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 108 T + 13254 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 320 T + 47934 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 236 T + 61982 T^{2} - 236 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 60 T + 8462 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 204 T + 108830 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 44 T + 106326 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 136 T + 81718 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 400 T + 106526 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 16 T + 260838 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 464 T + 458102 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 684 T + 535646 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 736 T + 602470 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 740 T + 757502 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 424 T + 748558 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 408 T - 143586 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 608 T + 1200710 T^{2} - 608 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1332 T + 1790774 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 2448 T + 3286542 T^{2} - 2448 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.031090576641441373614635692617, −8.845604869942392620376622277370, −8.514297371156935355032439595201, −8.474341455024723470936567120092, −7.68791290025835345030087649929, −7.54579351967256139237609065868, −6.95784187454072987476040509442, −6.34629257629547888247689509972, −6.20913155857813730264677779498, −5.91615389287345527139436811812, −4.79154641553107522513901292180, −4.76087308741234609122526767407, −4.13148405319743605170495231426, −3.92538522091243805615370411817, −3.40903490882162041523356912234, −2.34652686206042097081089699799, −1.85504130551169329715287371316, −1.61170418332528786560906348292, −0.815818025496236676430602633103, −0.39219303726823274276798959071, 0.39219303726823274276798959071, 0.815818025496236676430602633103, 1.61170418332528786560906348292, 1.85504130551169329715287371316, 2.34652686206042097081089699799, 3.40903490882162041523356912234, 3.92538522091243805615370411817, 4.13148405319743605170495231426, 4.76087308741234609122526767407, 4.79154641553107522513901292180, 5.91615389287345527139436811812, 6.20913155857813730264677779498, 6.34629257629547888247689509972, 6.95784187454072987476040509442, 7.54579351967256139237609065868, 7.68791290025835345030087649929, 8.474341455024723470936567120092, 8.514297371156935355032439595201, 8.845604869942392620376622277370, 9.031090576641441373614635692617

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