Properties

Label 4-1815e2-1.1-c1e2-0-31
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 5-s + 9-s + 4·12-s + 2·15-s + 2·20-s − 13·23-s − 4·25-s + 4·27-s − 10·31-s − 2·36-s − 6·37-s − 45-s + 7·47-s − 2·49-s + 14·53-s + 9·59-s − 4·60-s + 8·64-s + 10·67-s + 26·69-s − 3·71-s + 8·75-s − 11·81-s − 15·89-s + 26·92-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 0.447·5-s + 1/3·9-s + 1.15·12-s + 0.516·15-s + 0.447·20-s − 2.71·23-s − 4/5·25-s + 0.769·27-s − 1.79·31-s − 1/3·36-s − 0.986·37-s − 0.149·45-s + 1.02·47-s − 2/7·49-s + 1.92·53-s + 1.17·59-s − 0.516·60-s + 64-s + 1.22·67-s + 3.13·69-s − 0.356·71-s + 0.923·75-s − 1.22·81-s − 1.58·89-s + 2.71·92-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: \(2\)
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$C_2$ \( 1 + 2 T + p T^{2} \)
5$C_2$ \( 1 + T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
29$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.04502838324138087262290742522, −6.65762051109697230207565543540, −6.18403517660781390542273208435, −5.65444849394794378272019115383, −5.39455959902491503973853144411, −5.25814444227930818243728717556, −4.37556314110803754461332293634, −4.12641506336196742819547078348, −3.87311776450695472561956997807, −3.34454435137950146335564399019, −2.33616056309322288369627632674, −2.02164582772337007198218623263, −1.05051773239065976697308404272, 0, 0, 1.05051773239065976697308404272, 2.02164582772337007198218623263, 2.33616056309322288369627632674, 3.34454435137950146335564399019, 3.87311776450695472561956997807, 4.12641506336196742819547078348, 4.37556314110803754461332293634, 5.25814444227930818243728717556, 5.39455959902491503973853144411, 5.65444849394794378272019115383, 6.18403517660781390542273208435, 6.65762051109697230207565543540, 7.04502838324138087262290742522

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