Properties

Label 4-1815e2-1.1-c1e2-0-39
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  + 2·3-s + 3·4-s − 5-s + 9-s + 6·12-s − 2·15-s + 5·16-s − 3·20-s + 4·23-s − 4·25-s − 4·27-s − 4·31-s + 3·36-s − 45-s − 4·47-s + 10·48-s − 10·49-s − 18·53-s − 6·60-s + 3·64-s + 8·69-s − 8·75-s − 5·80-s − 11·81-s + 12·92-s − 8·93-s − 12·100-s + ⋯
L(s)  = 1  + 1.15·3-s + 3/2·4-s − 0.447·5-s + 1/3·9-s + 1.73·12-s − 0.516·15-s + 5/4·16-s − 0.670·20-s + 0.834·23-s − 4/5·25-s − 0.769·27-s − 0.718·31-s + 1/2·36-s − 0.149·45-s − 0.583·47-s + 1.44·48-s − 1.42·49-s − 2.47·53-s − 0.774·60-s + 3/8·64-s + 0.963·69-s − 0.923·75-s − 0.559·80-s − 1.22·81-s + 1.25·92-s − 0.829·93-s − 6/5·100-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: Trivial
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$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_2$ \( 1 + T + p T^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
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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

−7.40427146133772288890603425285, −6.95194937408196545169647872636, −6.64026482633843057610991692813, −6.06430677419634410533897008365, −5.88778555166678825622919487435, −5.13568217343320877230000855919, −4.77759963971963506796260245439, −4.13012524511606510335925926498, −3.54266250258779507084362046373, −3.26877703467396372279013603086, −2.85360060548728313366058099481, −2.33158054483474316617550778552, −1.78816646270829457744491884263, −1.37251404539753680780946795965, 0, 1.37251404539753680780946795965, 1.78816646270829457744491884263, 2.33158054483474316617550778552, 2.85360060548728313366058099481, 3.26877703467396372279013603086, 3.54266250258779507084362046373, 4.13012524511606510335925926498, 4.77759963971963506796260245439, 5.13568217343320877230000855919, 5.88778555166678825622919487435, 6.06430677419634410533897008365, 6.64026482633843057610991692813, 6.95194937408196545169647872636, 7.40427146133772288890603425285

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