Properties

Label 4-8624e2-1.1-c1e2-0-16
Degree $4$
Conductor $74373376$
Sign $1$
Analytic cond. $4742.11$
Root an. cond. $8.29837$
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  − 4·9-s + 2·11-s + 4·23-s − 8·25-s − 12·29-s − 4·37-s − 8·43-s − 24·53-s + 4·67-s + 20·71-s + 16·79-s + 7·81-s − 8·99-s + 40·107-s + 20·109-s − 36·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + ⋯
L(s)  = 1  − 4/3·9-s + 0.603·11-s + 0.834·23-s − 8/5·25-s − 2.22·29-s − 0.657·37-s − 1.21·43-s − 3.29·53-s + 0.488·67-s + 2.37·71-s + 1.80·79-s + 7/9·81-s − 0.804·99-s + 3.86·107-s + 1.91·109-s − 3.38·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74373376\)    =    \(2^{8} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4742.11\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 74373376,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \)
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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

−7.57876736028286376065700324215, −7.48459730219587332414766404185, −6.73290684429592846207519394460, −6.58457650196766177452272450964, −6.19857826851861701516336484992, −5.99605172076279026676006438248, −5.43056530756312923324311392347, −5.22093521201904546230196757482, −4.95343423736091596947228775424, −4.50153256994148336194726783437, −3.78276805804753880596918911825, −3.71828751442179370619937397480, −3.32145250485866782244814033173, −3.02267128261271388800815658486, −2.17502743322467007387981342736, −2.16070168030589674349300478711, −1.58126441160005025325976768153, −0.990126139818806984030646148806, 0, 0, 0.990126139818806984030646148806, 1.58126441160005025325976768153, 2.16070168030589674349300478711, 2.17502743322467007387981342736, 3.02267128261271388800815658486, 3.32145250485866782244814033173, 3.71828751442179370619937397480, 3.78276805804753880596918911825, 4.50153256994148336194726783437, 4.95343423736091596947228775424, 5.22093521201904546230196757482, 5.43056530756312923324311392347, 5.99605172076279026676006438248, 6.19857826851861701516336484992, 6.58457650196766177452272450964, 6.73290684429592846207519394460, 7.48459730219587332414766404185, 7.57876736028286376065700324215

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