Properties

Label 4-2475e2-1.1-c0e2-0-0
Degree $4$
Conductor $6125625$
Sign $1$
Analytic cond. $1.52568$
Root an. cond. $1.11138$
Motivic weight $0$
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  − 2·11-s − 16-s + 4·89-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 2·11-s − 16-s + 4·89-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6125625\)    =    \(3^{4} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.52568\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6125625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8293598445\)
\(L(\frac12)\) \(\approx\) \(0.8293598445\)
\(L(1)\) 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 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$ \( ( 1 - T )^{4} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−9.255446755635192883616413321123, −8.796382487019720156064712316084, −8.665890775692716984870400346682, −8.008697454278127680228133237409, −7.86787089806435487365283782211, −7.42725775038231072163251504412, −7.11162967738839176791672166010, −6.64864046221455927308715424955, −6.07882341212895919678951994991, −5.94928391167267172541956313069, −5.24750873227289428694305017670, −4.95013718549775742299814285568, −4.76344684937300545934072862640, −4.12775050035510841839462947434, −3.59593173238698278212631421531, −3.12705762268618250485070417587, −2.53491261925341392823887334934, −2.29292202697258242983783521447, −1.66272384420624847595547089639, −0.58782236177049019704470515032, 0.58782236177049019704470515032, 1.66272384420624847595547089639, 2.29292202697258242983783521447, 2.53491261925341392823887334934, 3.12705762268618250485070417587, 3.59593173238698278212631421531, 4.12775050035510841839462947434, 4.76344684937300545934072862640, 4.95013718549775742299814285568, 5.24750873227289428694305017670, 5.94928391167267172541956313069, 6.07882341212895919678951994991, 6.64864046221455927308715424955, 7.11162967738839176791672166010, 7.42725775038231072163251504412, 7.86787089806435487365283782211, 8.008697454278127680228133237409, 8.665890775692716984870400346682, 8.796382487019720156064712316084, 9.255446755635192883616413321123

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