Properties

Label 4-1440e2-1.1-c1e2-0-15
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 11-s + 4·17-s − 3·19-s − 25-s − 27-s + 33-s + 17·41-s + 15·43-s − 3·49-s − 4·51-s + 3·57-s + 9·59-s − 8·67-s + 8·73-s + 75-s + 81-s − 13·83-s + 6·89-s + 3·97-s − 99-s − 29·107-s − 8·113-s − 15·121-s − 17·123-s + 127-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.301·11-s + 0.970·17-s − 0.688·19-s − 1/5·25-s − 0.192·27-s + 0.174·33-s + 2.65·41-s + 2.28·43-s − 3/7·49-s − 0.560·51-s + 0.397·57-s + 1.17·59-s − 0.977·67-s + 0.936·73-s + 0.115·75-s + 1/9·81-s − 1.42·83-s + 0.635·89-s + 0.304·97-s − 0.100·99-s − 2.80·107-s − 0.752·113-s − 1.36·121-s − 1.53·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.681175389\)
\(L(\frac12)\) \(\approx\) \(1.681175389\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( 1 + T \)
5$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \)
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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

−7.69773387349558445940849340795, −7.40531887092939634297472250499, −6.83337841856452265832271990973, −6.48101245173312557658039811820, −5.87915255478957851899487299043, −5.63889078744194722339106756685, −5.37128456776400282815001675086, −4.58350473613441350124080584235, −4.23425676012761593597622140740, −3.92854119874088503774401256884, −3.14772016003112301690455206284, −2.62826339412341470417534805749, −2.11861300262478951845679348030, −1.24554361587952458581669196081, −0.59992592686626525677513952069, 0.59992592686626525677513952069, 1.24554361587952458581669196081, 2.11861300262478951845679348030, 2.62826339412341470417534805749, 3.14772016003112301690455206284, 3.92854119874088503774401256884, 4.23425676012761593597622140740, 4.58350473613441350124080584235, 5.37128456776400282815001675086, 5.63889078744194722339106756685, 5.87915255478957851899487299043, 6.48101245173312557658039811820, 6.83337841856452265832271990973, 7.40531887092939634297472250499, 7.69773387349558445940849340795

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