Properties

Label 4-903168-1.1-c1e2-0-24
Degree $4$
Conductor $903168$
Sign $-1$
Analytic cond. $57.5867$
Root an. cond. $2.75474$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 8·11-s − 4·17-s − 8·19-s − 6·25-s − 4·27-s − 16·33-s − 4·41-s − 16·43-s + 49-s + 8·51-s + 16·57-s + 24·59-s + 16·67-s + 4·73-s + 12·75-s + 5·81-s − 24·83-s + 12·89-s + 4·97-s + 24·99-s + 24·107-s − 28·113-s + 26·121-s + 8·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 2.41·11-s − 0.970·17-s − 1.83·19-s − 6/5·25-s − 0.769·27-s − 2.78·33-s − 0.624·41-s − 2.43·43-s + 1/7·49-s + 1.12·51-s + 2.11·57-s + 3.12·59-s + 1.95·67-s + 0.468·73-s + 1.38·75-s + 5/9·81-s − 2.63·83-s + 1.27·89-s + 0.406·97-s + 2.41·99-s + 2.32·107-s − 2.63·113-s + 2.36·121-s + 0.721·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(903168\)    =    \(2^{11} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(57.5867\)
Root analytic conductor: \(2.75474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 903168,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.190173500954779681525780731187, −7.20214522889553119059062687435, −6.78044077445841975719051191645, −6.66339990127808429560142628788, −6.35837082667327210170937090345, −5.74243037997109166467332435643, −5.36857771747617040863256466681, −4.52969287471330622027217048773, −4.43219841019389924830737127106, −3.69109001567993476839628496737, −3.61417461907435968748262507791, −2.19510980306732737447197047942, −1.90612310330978239121510440566, −1.06346104514984424021583213227, 0, 1.06346104514984424021583213227, 1.90612310330978239121510440566, 2.19510980306732737447197047942, 3.61417461907435968748262507791, 3.69109001567993476839628496737, 4.43219841019389924830737127106, 4.52969287471330622027217048773, 5.36857771747617040863256466681, 5.74243037997109166467332435643, 6.35837082667327210170937090345, 6.66339990127808429560142628788, 6.78044077445841975719051191645, 7.20214522889553119059062687435, 8.190173500954779681525780731187

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