Properties

Label 4-315e2-1.1-c1e2-0-6
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s − 8·8-s + 8·11-s − 7·16-s + 16·22-s + 25-s + 4·29-s + 14·32-s − 20·37-s + 8·43-s − 8·44-s − 7·49-s + 2·50-s + 20·53-s + 8·58-s + 35·64-s + 24·67-s + 16·71-s − 40·74-s + 16·86-s − 64·88-s − 14·98-s − 100-s + 40·106-s + 24·107-s + 28·109-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s + 2.41·11-s − 7/4·16-s + 3.41·22-s + 1/5·25-s + 0.742·29-s + 2.47·32-s − 3.28·37-s + 1.21·43-s − 1.20·44-s − 49-s + 0.282·50-s + 2.74·53-s + 1.05·58-s + 35/8·64-s + 2.93·67-s + 1.89·71-s − 4.64·74-s + 1.72·86-s − 6.82·88-s − 1.41·98-s − 0.0999·100-s + 3.88·106-s + 2.32·107-s + 2.68·109-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: $\chi_{99225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 99225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.253375518\)
\(L(\frac12)\) \(\approx\) \(2.253375518\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.730283045024072287400674871237, −8.890589428908360265391160029852, −8.627000477337622059216224790867, −8.554588868470699894187777077269, −7.37485778338999061621083536758, −6.76955885158912340897756950945, −6.39356342307905775022074491331, −5.91397914498304472267826736485, −5.04860090093668311608273372623, −5.03119182641673168471146566272, −4.09922216243454775532088496022, −3.63412818236731287070311550143, −3.56571005686312005410318267112, −2.32940333880865099555943992805, −0.964820104627525128067217267144, 0.964820104627525128067217267144, 2.32940333880865099555943992805, 3.56571005686312005410318267112, 3.63412818236731287070311550143, 4.09922216243454775532088496022, 5.03119182641673168471146566272, 5.04860090093668311608273372623, 5.91397914498304472267826736485, 6.39356342307905775022074491331, 6.76955885158912340897756950945, 7.37485778338999061621083536758, 8.554588868470699894187777077269, 8.627000477337622059216224790867, 8.890589428908360265391160029852, 9.730283045024072287400674871237

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