Properties

Label 4-1386e2-1.1-c1e2-0-27
Degree $4$
Conductor $1920996$
Sign $1$
Analytic cond. $122.484$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·5-s − 5·7-s + 8-s + 4·10-s − 11-s − 2·13-s + 5·14-s − 16-s + 2·17-s − 6·19-s + 22-s − 2·23-s + 5·25-s + 2·26-s − 2·29-s − 4·31-s − 2·34-s + 20·35-s + 2·37-s + 6·38-s − 4·40-s + 4·41-s + 8·43-s + 2·46-s + 2·47-s + 18·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.78·5-s − 1.88·7-s + 0.353·8-s + 1.26·10-s − 0.301·11-s − 0.554·13-s + 1.33·14-s − 1/4·16-s + 0.485·17-s − 1.37·19-s + 0.213·22-s − 0.417·23-s + 25-s + 0.392·26-s − 0.371·29-s − 0.718·31-s − 0.342·34-s + 3.38·35-s + 0.328·37-s + 0.973·38-s − 0.632·40-s + 0.624·41-s + 1.21·43-s + 0.294·46-s + 0.291·47-s + 18/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1920996\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(122.484\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1386} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1920996,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
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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

−9.502920509103336239377245719260, −9.201647939686874887845029844992, −8.381601595198172497295150491774, −8.237682208326296493849964574679, −7.72911967490789593129587584884, −7.54603717305674134171837986224, −7.01367227274302486257420815689, −6.62760607304414910483891663431, −6.27751924547485755292347136030, −5.66563171531911962519381403483, −5.25034017071322233503876085325, −4.49395480179660569319583333273, −4.05122305797424573947698127282, −3.75101847876993285462280275100, −3.41286312951143572308018911181, −2.57653131589788772638924663908, −2.29754830802319300278492286719, −1.00503494855272045806181227706, 0, 0, 1.00503494855272045806181227706, 2.29754830802319300278492286719, 2.57653131589788772638924663908, 3.41286312951143572308018911181, 3.75101847876993285462280275100, 4.05122305797424573947698127282, 4.49395480179660569319583333273, 5.25034017071322233503876085325, 5.66563171531911962519381403483, 6.27751924547485755292347136030, 6.62760607304414910483891663431, 7.01367227274302486257420815689, 7.54603717305674134171837986224, 7.72911967490789593129587584884, 8.237682208326296493849964574679, 8.381601595198172497295150491774, 9.201647939686874887845029844992, 9.502920509103336239377245719260

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