Properties

Label 4-315e2-1.1-c1e2-0-29
Degree $4$
Conductor $99225$
Sign $1$
Analytic cond. $6.32667$
Root an. cond. $1.58596$
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  − 3·3-s − 2·4-s − 5-s − 5·7-s + 6·9-s + 3·11-s + 6·12-s + 3·15-s − 12·17-s + 2·20-s + 15·21-s + 6·23-s − 9·27-s + 10·28-s − 15·29-s + 6·31-s − 9·33-s + 5·35-s − 12·36-s − 16·37-s − 6·41-s − 10·43-s − 6·44-s − 6·45-s − 9·47-s + 18·49-s + 36·51-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 0.447·5-s − 1.88·7-s + 2·9-s + 0.904·11-s + 1.73·12-s + 0.774·15-s − 2.91·17-s + 0.447·20-s + 3.27·21-s + 1.25·23-s − 1.73·27-s + 1.88·28-s − 2.78·29-s + 1.07·31-s − 1.56·33-s + 0.845·35-s − 2·36-s − 2.63·37-s − 0.937·41-s − 1.52·43-s − 0.904·44-s − 0.894·45-s − 1.31·47-s + 18/7·49-s + 5.04·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(99225\)    =    \(3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(6.32667\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 99225,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 139 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−11.32676804656343459989476409285, −11.11354614702334595031068555394, −10.56051525568227247938542916333, −10.11207876206754506228091195866, −9.359240673800704858865599609601, −9.331098130362485228446961979221, −8.820354168012346425549062737348, −8.278196322979109155403078955892, −7.05289880414395618496727379726, −6.81301939619650145705207178865, −6.72535267588907873777839382021, −6.13531485755614738198360963806, −5.16626329699123585410935558924, −5.13231194078843606176249244297, −4.06201787591128922717530840258, −4.04357415347202830850886672646, −3.15037463265325187270500173693, −1.78197123018775952764122594914, 0, 0, 1.78197123018775952764122594914, 3.15037463265325187270500173693, 4.04357415347202830850886672646, 4.06201787591128922717530840258, 5.13231194078843606176249244297, 5.16626329699123585410935558924, 6.13531485755614738198360963806, 6.72535267588907873777839382021, 6.81301939619650145705207178865, 7.05289880414395618496727379726, 8.278196322979109155403078955892, 8.820354168012346425549062737348, 9.331098130362485228446961979221, 9.359240673800704858865599609601, 10.11207876206754506228091195866, 10.56051525568227247938542916333, 11.11354614702334595031068555394, 11.32676804656343459989476409285

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