Properties

Degree $4$
Conductor $14745600$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s + 3·9-s + 4·13-s − 4·15-s − 25-s − 4·27-s − 20·37-s − 8·39-s − 4·41-s − 24·43-s + 6·45-s + 10·49-s − 20·53-s + 8·65-s − 16·67-s − 8·71-s + 2·75-s − 16·79-s + 5·81-s − 8·83-s + 12·89-s − 8·107-s + 40·111-s + 12·117-s + 18·121-s + 8·123-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 9-s + 1.10·13-s − 1.03·15-s − 1/5·25-s − 0.769·27-s − 3.28·37-s − 1.28·39-s − 0.624·41-s − 3.65·43-s + 0.894·45-s + 10/7·49-s − 2.74·53-s + 0.992·65-s − 1.95·67-s − 0.949·71-s + 0.230·75-s − 1.80·79-s + 5/9·81-s − 0.878·83-s + 1.27·89-s − 0.773·107-s + 3.79·111-s + 1.10·117-s + 1.63·121-s + 0.721·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4651954439\)
\(L(\frac12)\) \(\approx\) \(0.4651954439\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.561753271073022128325897227349, −8.449617658935682980101228900197, −8.054583092796113678871690138821, −7.26665600984162297707491558447, −7.13189537479291351882587072422, −6.85242489540295407919268452300, −6.23523162061230787235868153417, −6.09899961831971042395416080145, −5.85395243289108653291132987370, −5.25039049660556180526269115043, −4.98719865974542257406460526183, −4.78381778627784541015896606327, −4.11091320673838646542904295556, −3.61461112464794846180438201666, −3.28607537413914765863373658962, −2.84451697234790793650844228161, −1.78773574803789198224948813380, −1.70187213293398571956481567266, −1.37645479050869831430177493519, −0.20946061459046152535788349015, 0.20946061459046152535788349015, 1.37645479050869831430177493519, 1.70187213293398571956481567266, 1.78773574803789198224948813380, 2.84451697234790793650844228161, 3.28607537413914765863373658962, 3.61461112464794846180438201666, 4.11091320673838646542904295556, 4.78381778627784541015896606327, 4.98719865974542257406460526183, 5.25039049660556180526269115043, 5.85395243289108653291132987370, 6.09899961831971042395416080145, 6.23523162061230787235868153417, 6.85242489540295407919268452300, 7.13189537479291351882587072422, 7.26665600984162297707491558447, 8.054583092796113678871690138821, 8.449617658935682980101228900197, 8.561753271073022128325897227349

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