Properties

Label 4-1815e2-1.1-c1e2-0-32
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 + 2·4-s − 3·5-s + 9-s + 4·12-s − 6·15-s − 6·20-s + 9·23-s + 4·25-s − 4·27-s + 2·31-s + 2·36-s + 2·37-s − 3·45-s − 3·47-s − 2·49-s + 6·53-s − 15·59-s − 12·60-s − 8·64-s − 4·67-s + 18·69-s − 21·71-s + 8·75-s − 11·81-s + 9·89-s + 18·92-s + ⋯
L(s)  = 1  + 1.15·3-s + 4-s − 1.34·5-s + 1/3·9-s + 1.15·12-s − 1.54·15-s − 1.34·20-s + 1.87·23-s + 4/5·25-s − 0.769·27-s + 0.359·31-s + 1/3·36-s + 0.328·37-s − 0.447·45-s − 0.437·47-s − 2/7·49-s + 0.824·53-s − 1.95·59-s − 1.54·60-s − 64-s − 0.488·67-s + 2.16·69-s − 2.49·71-s + 0.923·75-s − 1.22·81-s + 0.953·89-s + 1.87·92-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: \(1\)
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$C_2$ \( 1 + 3 T + p T^{2} \)
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$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 88 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 97 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.34756777576650051168982575516, −7.09151703389172071570779375970, −6.69262788913065898083037154346, −6.06793432993445778961105442807, −5.85264403529183768793985571586, −4.95142067594325456354149414745, −4.72474810492383340680468051662, −4.21634777490546200581758944721, −3.62745758228872065953096497687, −3.24553198791295731657302119183, −2.85876087849156628093692285308, −2.50720547629111837091247746292, −1.76876553601419242309310801336, −1.12894237257338866341798982582, 0, 1.12894237257338866341798982582, 1.76876553601419242309310801336, 2.50720547629111837091247746292, 2.85876087849156628093692285308, 3.24553198791295731657302119183, 3.62745758228872065953096497687, 4.21634777490546200581758944721, 4.72474810492383340680468051662, 4.95142067594325456354149414745, 5.85264403529183768793985571586, 6.06793432993445778961105442807, 6.69262788913065898083037154346, 7.09151703389172071570779375970, 7.34756777576650051168982575516

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