Properties

Label 4-165e2-1.1-c1e2-0-12
Degree $4$
Conductor $27225$
Sign $-1$
Analytic cond. $1.73588$
Root an. cond. $1.14783$
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  − 4-s − 2·5-s − 9-s − 3·16-s − 8·19-s + 2·20-s − 25-s + 36-s + 2·45-s − 6·49-s − 16·59-s − 12·61-s + 7·64-s + 8·71-s + 8·76-s + 6·80-s + 81-s + 4·89-s + 16·95-s + 100-s − 4·109-s + 121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 1/3·9-s − 3/4·16-s − 1.83·19-s + 0.447·20-s − 1/5·25-s + 1/6·36-s + 0.298·45-s − 6/7·49-s − 2.08·59-s − 1.53·61-s + 7/8·64-s + 0.949·71-s + 0.917·76-s + 0.670·80-s + 1/9·81-s + 0.423·89-s + 1.64·95-s + 1/10·100-s − 0.383·109-s + 1/11·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1.73588\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 27225,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−10.53139051654663073555070393319, −9.659040628966221382673419857376, −9.270314714382499031583985283302, −8.651071963733801398684051422985, −8.263258458125184732067630125524, −7.75530179739035787583056267962, −7.09800681503408718898952654169, −6.40034752089315802554982876457, −5.97060718272721344317182488671, −4.94140996552388691383319894400, −4.48181160467337111285268927845, −3.91438791397988646765919874549, −3.07004798748965751972422130953, −2.00308110898687829211187295156, 0, 2.00308110898687829211187295156, 3.07004798748965751972422130953, 3.91438791397988646765919874549, 4.48181160467337111285268927845, 4.94140996552388691383319894400, 5.97060718272721344317182488671, 6.40034752089315802554982876457, 7.09800681503408718898952654169, 7.75530179739035787583056267962, 8.263258458125184732067630125524, 8.651071963733801398684051422985, 9.270314714382499031583985283302, 9.659040628966221382673419857376, 10.53139051654663073555070393319

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