Properties

Label 4-1575e2-1.1-c0e2-0-2
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $0.617839$
Root an. cond. $0.886581$
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  + 4-s + 2·11-s − 2·29-s + 2·44-s − 49-s − 64-s + 2·71-s + 2·79-s + 2·109-s − 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 196-s + ⋯
L(s)  = 1  + 4-s + 2·11-s − 2·29-s + 2·44-s − 49-s − 64-s + 2·71-s + 2·79-s + 2·109-s − 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 196-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.617839\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1575} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2480625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.623019609\)
\(L(\frac12)\) \(\approx\) \(1.623019609\)
\(L(1)\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2^2$ \( 1 - T^{2} + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 - T^{2} + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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

−9.697831863137748353464176588330, −9.477065285462139530277909108834, −8.976846395961361110782148213354, −8.808036408677867291682335091912, −8.205357190835298406048859052435, −7.66476373088840608831281575575, −7.42557907845317365135588293843, −6.96840610634510930146297951669, −6.52929929537319611138161596603, −6.27966694123544293748218645752, −6.01379839869926721671148843739, −5.21546728565667259673098134671, −5.00286332828853934429028089546, −4.20615326396593357826942978377, −3.82671895408862407941536656031, −3.51480041582539475256755452809, −2.87465698894886530916011711200, −2.08086484378021364477569432625, −1.84764916556910467273797050135, −1.09915149154151201136701894336, 1.09915149154151201136701894336, 1.84764916556910467273797050135, 2.08086484378021364477569432625, 2.87465698894886530916011711200, 3.51480041582539475256755452809, 3.82671895408862407941536656031, 4.20615326396593357826942978377, 5.00286332828853934429028089546, 5.21546728565667259673098134671, 6.01379839869926721671148843739, 6.27966694123544293748218645752, 6.52929929537319611138161596603, 6.96840610634510930146297951669, 7.42557907845317365135588293843, 7.66476373088840608831281575575, 8.205357190835298406048859052435, 8.808036408677867291682335091912, 8.976846395961361110782148213354, 9.477065285462139530277909108834, 9.697831863137748353464176588330

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