Properties

Label 4-1110e2-1.1-c1e2-0-23
Degree $4$
Conductor $1232100$
Sign $-1$
Analytic cond. $78.5597$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s + 2·7-s + 9-s + 2·12-s + 16-s − 12·19-s + 4·21-s + 25-s − 4·27-s + 2·28-s + 6·31-s + 36-s − 2·37-s − 2·43-s + 2·48-s − 11·49-s − 24·57-s − 30·61-s + 2·63-s + 64-s + 2·75-s − 12·76-s − 16·79-s − 11·81-s + 4·84-s + 12·93-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.577·12-s + 1/4·16-s − 2.75·19-s + 0.872·21-s + 1/5·25-s − 0.769·27-s + 0.377·28-s + 1.07·31-s + 1/6·36-s − 0.328·37-s − 0.304·43-s + 0.288·48-s − 1.57·49-s − 3.17·57-s − 3.84·61-s + 0.251·63-s + 1/8·64-s + 0.230·75-s − 1.37·76-s − 1.80·79-s − 1.22·81-s + 0.436·84-s + 1.24·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1232100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(78.5597\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1232100} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1232100,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
37$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−7.947480595907964918208604648618, −7.55947946358028496449425566831, −6.89931657836489782864864420449, −6.56405990058932844390013504676, −6.08056229568137165407273954903, −5.77272193873314481326296907642, −4.83332190202465636941309325541, −4.63919100492136925221940798677, −4.12586421650220620613319696777, −3.56154188220255317605428378165, −2.82505147340032154850850367219, −2.65455170884079945670816110578, −1.73957054101312727200784638700, −1.63805431218632083963465992264, 0, 1.63805431218632083963465992264, 1.73957054101312727200784638700, 2.65455170884079945670816110578, 2.82505147340032154850850367219, 3.56154188220255317605428378165, 4.12586421650220620613319696777, 4.63919100492136925221940798677, 4.83332190202465636941309325541, 5.77272193873314481326296907642, 6.08056229568137165407273954903, 6.56405990058932844390013504676, 6.89931657836489782864864420449, 7.55947946358028496449425566831, 7.947480595907964918208604648618

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