Properties

Label 4-1815e2-1.1-c1e2-0-28
Degree $4$
Conductor $3294225$
Sign $-1$
Analytic cond. $210.042$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 2·9-s + 15-s − 4·16-s + 2·23-s − 4·25-s − 5·27-s + 14·31-s − 2·45-s + 16·47-s − 4·48-s − 10·49-s − 12·53-s + 2·69-s − 4·75-s − 4·80-s + 81-s + 14·93-s − 18·113-s + 2·115-s − 9·125-s + 127-s + 131-s − 5·135-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.258·15-s − 16-s + 0.417·23-s − 4/5·25-s − 0.962·27-s + 2.51·31-s − 0.298·45-s + 2.33·47-s − 0.577·48-s − 1.42·49-s − 1.64·53-s + 0.240·69-s − 0.461·75-s − 0.447·80-s + 1/9·81-s + 1.45·93-s − 1.69·113-s + 0.186·115-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s − 0.430·135-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(210.042\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3294225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 3294225,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−7.39893594642771643206543547031, −6.80044609023638042209450417099, −6.47231251560475429244143852591, −6.09966792552373038662002103453, −5.73870626081545475442824548628, −5.13615482341982682420909731232, −4.77016036292615901681122983244, −4.30413573577731527020603612370, −3.82123745760857088121774501968, −3.21993795263974952405435891110, −2.67401217228157830548156869283, −2.46478514923094428626030938829, −1.78907311686180992193673629119, −1.03757175929891373992477223583, 0, 1.03757175929891373992477223583, 1.78907311686180992193673629119, 2.46478514923094428626030938829, 2.67401217228157830548156869283, 3.21993795263974952405435891110, 3.82123745760857088121774501968, 4.30413573577731527020603612370, 4.77016036292615901681122983244, 5.13615482341982682420909731232, 5.73870626081545475442824548628, 6.09966792552373038662002103453, 6.47231251560475429244143852591, 6.80044609023638042209450417099, 7.39893594642771643206543547031

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