Properties

Label 4-1575e2-1.1-c1e2-0-7
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
Motivic weight $1$
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  − 2·2-s + 4-s + 2·7-s + 4·11-s + 4·13-s − 4·14-s + 16-s − 4·17-s − 8·22-s − 4·23-s − 8·26-s + 2·28-s + 16·29-s + 2·32-s + 8·34-s + 12·37-s − 4·41-s + 8·43-s + 4·44-s + 8·46-s + 8·47-s + 3·49-s + 4·52-s − 16·53-s − 32·58-s − 8·59-s + 12·61-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.755·7-s + 1.20·11-s + 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 1.70·22-s − 0.834·23-s − 1.56·26-s + 0.377·28-s + 2.97·29-s + 0.353·32-s + 1.37·34-s + 1.97·37-s − 0.624·41-s + 1.21·43-s + 0.603·44-s + 1.17·46-s + 1.16·47-s + 3/7·49-s + 0.554·52-s − 2.19·53-s − 4.20·58-s − 1.04·59-s + 1.53·61-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.350298877\)
\(L(\frac12)\) \(\approx\) \(1.350298877\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−9.285349067717175880142989476131, −9.281241068477553963774117269923, −8.728426403602027105611871205858, −8.683523398064245931358573386170, −8.112904435866549713564142443748, −7.76810097449512895131178544168, −7.62873576403561617131577714023, −6.63093532373398252874187639222, −6.42487303242401377188897156611, −6.31858370100240807587918287960, −5.73240979539694296258335927157, −4.94815588590116147010231314792, −4.51655591641503877926536922441, −4.28872564812333637304099569407, −3.67706109614843316059969298026, −3.06720417886087042793904368961, −2.38002570622359334650757213469, −1.83627792410151497358758155052, −0.891019597558121091142297421618, −0.864988085239673802908259165882, 0.864988085239673802908259165882, 0.891019597558121091142297421618, 1.83627792410151497358758155052, 2.38002570622359334650757213469, 3.06720417886087042793904368961, 3.67706109614843316059969298026, 4.28872564812333637304099569407, 4.51655591641503877926536922441, 4.94815588590116147010231314792, 5.73240979539694296258335927157, 6.31858370100240807587918287960, 6.42487303242401377188897156611, 6.63093532373398252874187639222, 7.62873576403561617131577714023, 7.76810097449512895131178544168, 8.112904435866549713564142443748, 8.683523398064245931358573386170, 8.728426403602027105611871205858, 9.281241068477553963774117269923, 9.285349067717175880142989476131

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