Properties

Label 4-1575e2-1.1-c1e2-0-13
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·7-s − 3·8-s − 11-s − 5·13-s − 2·14-s + 16-s − 5·17-s − 6·19-s + 22-s − 2·23-s + 5·26-s + 2·28-s − 29-s + 32-s + 5·34-s − 12·37-s + 6·38-s − 2·41-s − 10·43-s − 44-s + 2·46-s − 5·47-s + 3·49-s − 5·52-s − 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 1.06·8-s − 0.301·11-s − 1.38·13-s − 0.534·14-s + 1/4·16-s − 1.21·17-s − 1.37·19-s + 0.213·22-s − 0.417·23-s + 0.980·26-s + 0.377·28-s − 0.185·29-s + 0.176·32-s + 0.857·34-s − 1.97·37-s + 0.973·38-s − 0.312·41-s − 1.52·43-s − 0.150·44-s + 0.294·46-s − 0.729·47-s + 3/7·49-s − 0.693·52-s − 0.274·53-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: \(2\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−8.974461626590085730877765775247, −8.857664306828943862619908537831, −8.512769786128803042405763142343, −8.214377903058339903323868904272, −7.62553134897065989058171359457, −7.27613823576566809276337895181, −6.86623343592287364109257264726, −6.42724403992767984401364677976, −6.22973029611827376351658369969, −5.46027936825311731931033431195, −4.98272862173393119618273505555, −4.82960689747777365542531811520, −4.16707225803529488355952294427, −3.66107307481191790408273410344, −2.97980200135367741907991651169, −2.31451230682140083030351424908, −2.15207966297178687611243312945, −1.48819737723106523844865324567, 0, 0, 1.48819737723106523844865324567, 2.15207966297178687611243312945, 2.31451230682140083030351424908, 2.97980200135367741907991651169, 3.66107307481191790408273410344, 4.16707225803529488355952294427, 4.82960689747777365542531811520, 4.98272862173393119618273505555, 5.46027936825311731931033431195, 6.22973029611827376351658369969, 6.42724403992767984401364677976, 6.86623343592287364109257264726, 7.27613823576566809276337895181, 7.62553134897065989058171359457, 8.214377903058339903323868904272, 8.512769786128803042405763142343, 8.857664306828943862619908537831, 8.974461626590085730877765775247

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