Properties

Label 4-2200e2-1.1-c1e2-0-18
Degree $4$
Conductor $4840000$
Sign $1$
Analytic cond. $308.602$
Root an. cond. $4.19131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s − 9-s − 2·11-s + 2·13-s − 4·17-s − 8·19-s − 2·21-s − 9·23-s − 2·29-s − 7·31-s + 2·33-s + 11·37-s − 2·39-s + 6·41-s + 6·43-s − 16·47-s + 6·49-s + 4·51-s − 8·53-s + 8·57-s − 5·59-s − 6·61-s − 2·63-s − 15·67-s + 9·69-s − 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s − 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.970·17-s − 1.83·19-s − 0.436·21-s − 1.87·23-s − 0.371·29-s − 1.25·31-s + 0.348·33-s + 1.80·37-s − 0.320·39-s + 0.937·41-s + 0.914·43-s − 2.33·47-s + 6/7·49-s + 0.560·51-s − 1.09·53-s + 1.05·57-s − 0.650·59-s − 0.768·61-s − 0.251·63-s − 1.83·67-s + 1.08·69-s − 0.593·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4840000\)    =    \(2^{6} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(308.602\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4840000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 15 T + 186 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 27 T + 372 T^{2} + 27 p T^{3} + p^{2} T^{4} \)
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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

−8.718490618376953769657621945176, −8.446064600014388309375931891414, −8.071214054493609649220467166396, −7.88784763572954475495488887979, −7.19376664471807480746460691025, −7.07837775051419882091553499224, −6.18266186323972847948873728683, −6.11840268150356585086802117767, −5.82359585408439039910600236430, −5.47666185808588562652194867158, −4.60767606098350367401776336190, −4.36757011990677979619389615173, −4.32085641328493261255085052798, −3.56328141390311271915817466986, −2.86992286077222977301869739858, −2.41191421436362608876075022357, −1.82365582511694199617454976320, −1.44965321046988511753828206052, 0, 0, 1.44965321046988511753828206052, 1.82365582511694199617454976320, 2.41191421436362608876075022357, 2.86992286077222977301869739858, 3.56328141390311271915817466986, 4.32085641328493261255085052798, 4.36757011990677979619389615173, 4.60767606098350367401776336190, 5.47666185808588562652194867158, 5.82359585408439039910600236430, 6.11840268150356585086802117767, 6.18266186323972847948873728683, 7.07837775051419882091553499224, 7.19376664471807480746460691025, 7.88784763572954475495488887979, 8.071214054493609649220467166396, 8.446064600014388309375931891414, 8.718490618376953769657621945176

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