Properties

Label 4-5520e2-1.1-c1e2-0-9
Degree $4$
Conductor $30470400$
Sign $1$
Analytic cond. $1942.81$
Root an. cond. $6.63908$
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  − 2·3-s + 2·5-s + 2·7-s + 3·9-s + 4·11-s − 8·13-s − 4·15-s − 6·17-s − 4·21-s + 2·23-s + 3·25-s − 4·27-s − 6·29-s + 10·31-s − 8·33-s + 4·35-s − 6·37-s + 16·39-s − 6·41-s − 4·43-s + 6·45-s − 8·47-s − 11·49-s + 12·51-s − 10·53-s + 8·55-s + 10·59-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 1.20·11-s − 2.21·13-s − 1.03·15-s − 1.45·17-s − 0.872·21-s + 0.417·23-s + 3/5·25-s − 0.769·27-s − 1.11·29-s + 1.79·31-s − 1.39·33-s + 0.676·35-s − 0.986·37-s + 2.56·39-s − 0.937·41-s − 0.609·43-s + 0.894·45-s − 1.16·47-s − 1.57·49-s + 1.68·51-s − 1.37·53-s + 1.07·55-s + 1.30·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30470400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1942.81\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 30470400,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 10 T + 55 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 6 T + 89 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 129 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 10 T + 93 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 12 T + 108 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 127 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 10 T + 117 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 96 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 234 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 202 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.74953599290933879962266125461, −7.71143626023053175143171229734, −6.99634598576095783817616187693, −6.81829885701393525496777589447, −6.47135976633997813671930328080, −6.46213577116685072095103128848, −5.73200606517624761887001746110, −5.40689523529663712690479257581, −5.01550542870153924814862024251, −4.80062121280810881805876208081, −4.41599738224735621597318883961, −4.27045187131788394301113708874, −3.39035498720159624891161968549, −3.06259212135660160400564661163, −2.38698678545669935520309868642, −2.03660463170726851993443952445, −1.48798733054522071923383153400, −1.29673460011563917580190857719, 0, 0, 1.29673460011563917580190857719, 1.48798733054522071923383153400, 2.03660463170726851993443952445, 2.38698678545669935520309868642, 3.06259212135660160400564661163, 3.39035498720159624891161968549, 4.27045187131788394301113708874, 4.41599738224735621597318883961, 4.80062121280810881805876208081, 5.01550542870153924814862024251, 5.40689523529663712690479257581, 5.73200606517624761887001746110, 6.46213577116685072095103128848, 6.47135976633997813671930328080, 6.81829885701393525496777589447, 6.99634598576095783817616187693, 7.71143626023053175143171229734, 7.74953599290933879962266125461

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