Properties

Label 4-1260e2-1.1-c1e2-0-13
Degree $4$
Conductor $1587600$
Sign $1$
Analytic cond. $101.226$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2·5-s + 10·13-s + 4·16-s − 6·17-s − 4·20-s + 3·25-s − 6·29-s + 4·37-s + 24·41-s + 49-s − 20·52-s − 24·53-s + 16·61-s − 8·64-s + 20·65-s + 12·68-s + 4·73-s + 8·80-s − 12·85-s + 24·89-s − 2·97-s − 6·100-s − 12·101-s − 14·109-s − 12·113-s + 12·116-s + ⋯
L(s)  = 1  − 4-s + 0.894·5-s + 2.77·13-s + 16-s − 1.45·17-s − 0.894·20-s + 3/5·25-s − 1.11·29-s + 0.657·37-s + 3.74·41-s + 1/7·49-s − 2.77·52-s − 3.29·53-s + 2.04·61-s − 64-s + 2.48·65-s + 1.45·68-s + 0.468·73-s + 0.894·80-s − 1.30·85-s + 2.54·89-s − 0.203·97-s − 3/5·100-s − 1.19·101-s − 1.34·109-s − 1.12·113-s + 1.11·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1587600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(101.226\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1587600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1587600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.324925606\)
\(L(\frac12)\) \(\approx\) \(2.324925606\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−8.087023964728641191414598204496, −7.67755521010834815156907872378, −6.69771158952637091109528432278, −6.60444721122988741193728663303, −6.06485042125071096226940610377, −5.63914839037632160871019437372, −5.54008340383010136479817620846, −4.54321060660836499326534703030, −4.41617775358459899843679315087, −3.82466491546191947603335262315, −3.45067056060882590557565434961, −2.72164418406817349261491719715, −2.03544754569172366495048437388, −1.35738749247200780270220785705, −0.72587888295584277318850692735, 0.72587888295584277318850692735, 1.35738749247200780270220785705, 2.03544754569172366495048437388, 2.72164418406817349261491719715, 3.45067056060882590557565434961, 3.82466491546191947603335262315, 4.41617775358459899843679315087, 4.54321060660836499326534703030, 5.54008340383010136479817620846, 5.63914839037632160871019437372, 6.06485042125071096226940610377, 6.60444721122988741193728663303, 6.69771158952637091109528432278, 7.67755521010834815156907872378, 8.087023964728641191414598204496

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