Properties

Label 4-1815e2-1.1-c1e2-0-12
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 − 3·4-s + 5-s − 2·9-s + 3·12-s − 15-s + 5·16-s − 3·20-s − 12·23-s − 4·25-s + 5·27-s − 31-s + 6·36-s + 6·37-s − 2·45-s + 10·47-s − 5·48-s + 12·49-s − 7·53-s + 12·59-s + 3·60-s − 3·64-s − 5·67-s + 12·69-s − 3·71-s + 4·75-s + 5·80-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 0.447·5-s − 2/3·9-s + 0.866·12-s − 0.258·15-s + 5/4·16-s − 0.670·20-s − 2.50·23-s − 4/5·25-s + 0.962·27-s − 0.179·31-s + 36-s + 0.986·37-s − 0.298·45-s + 1.45·47-s − 0.721·48-s + 12/7·49-s − 0.961·53-s + 1.56·59-s + 0.387·60-s − 3/8·64-s − 0.610·67-s + 1.44·69-s − 0.356·71-s + 0.461·75-s + 0.559·80-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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
61$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
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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

−7.41949878048906135622005174579, −6.78412443955357294813716306122, −6.29443683856812784768101205285, −5.87477767941063352231642757437, −5.62105117210868757879626010833, −5.38149760109488770530288296009, −4.75513180799144398252338231050, −4.21793669363561189835315566674, −4.02121820940130964332915653038, −3.60426081327829454842750009307, −2.69384369214365109075554586399, −2.35262794883262996468820257550, −1.55591218994608419117790081314, −0.67572510247103824890075449546, 0, 0.67572510247103824890075449546, 1.55591218994608419117790081314, 2.35262794883262996468820257550, 2.69384369214365109075554586399, 3.60426081327829454842750009307, 4.02121820940130964332915653038, 4.21793669363561189835315566674, 4.75513180799144398252338231050, 5.38149760109488770530288296009, 5.62105117210868757879626010833, 5.87477767941063352231642757437, 6.29443683856812784768101205285, 6.78412443955357294813716306122, 7.41949878048906135622005174579

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