Properties

Label 4-2200e2-1.1-c1e2-0-16
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 − 3·7-s − 9-s + 2·11-s − 2·13-s − 7·17-s + 9·19-s + 3·21-s − 6·23-s − 5·29-s + 5·31-s − 2·33-s + 37-s + 2·39-s − 20·41-s − 2·43-s + 6·47-s − 3·49-s + 7·51-s + 13·53-s − 9·57-s − 2·59-s − 3·61-s + 3·63-s + 6·69-s − 9·71-s − 18·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s − 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.69·17-s + 2.06·19-s + 0.654·21-s − 1.25·23-s − 0.928·29-s + 0.898·31-s − 0.348·33-s + 0.164·37-s + 0.320·39-s − 3.12·41-s − 0.304·43-s + 0.875·47-s − 3/7·49-s + 0.980·51-s + 1.78·53-s − 1.19·57-s − 0.260·59-s − 0.384·61-s + 0.377·63-s + 0.722·69-s − 1.06·71-s − 2.10·73-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$D_{4}$ \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_4$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 2 T - 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 120 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 9 T + 158 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 26 T + 346 T^{2} + 26 p T^{3} + p^{2} T^{4} \)
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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

−8.754326514318178105048230077046, −8.664844501141892481235139856392, −8.168690694091231830047867080055, −7.53835886051522739675359849153, −7.13212239517330728218967992933, −7.02098137938668714009051259072, −6.41118113912557275957496763110, −6.22068970411549740739863278979, −5.69398656639830046496126811810, −5.36260281926074019882784725267, −4.96670183560935242326581246944, −4.34960802384987719795808586581, −3.96971013751382362428356247847, −3.52029124049328955310852894781, −2.90828889456437385287581801841, −2.65312902819391403683713662019, −1.81062257481611373367775304370, −1.28905278132485924618995987126, 0, 0, 1.28905278132485924618995987126, 1.81062257481611373367775304370, 2.65312902819391403683713662019, 2.90828889456437385287581801841, 3.52029124049328955310852894781, 3.96971013751382362428356247847, 4.34960802384987719795808586581, 4.96670183560935242326581246944, 5.36260281926074019882784725267, 5.69398656639830046496126811810, 6.22068970411549740739863278979, 6.41118113912557275957496763110, 7.02098137938668714009051259072, 7.13212239517330728218967992933, 7.53835886051522739675359849153, 8.168690694091231830047867080055, 8.664844501141892481235139856392, 8.754326514318178105048230077046

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