Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{6} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s + 2·11-s − 9·13-s − 2·19-s + 5·23-s − 5·25-s + 15·29-s − 31-s − 2·35-s + 3·37-s − 19·41-s + 2·43-s − 3·47-s + 3·49-s − 53-s + 2·55-s − 8·59-s − 9·65-s + 19·67-s − 71-s − 6·73-s − 4·77-s + 13·79-s + 83-s − 15·89-s + 18·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.603·11-s − 2.49·13-s − 0.458·19-s + 1.04·23-s − 25-s + 2.78·29-s − 0.179·31-s − 0.338·35-s + 0.493·37-s − 2.96·41-s + 0.304·43-s − 0.437·47-s + 3/7·49-s − 0.137·53-s + 0.269·55-s − 1.04·59-s − 1.11·65-s + 2.32·67-s − 0.118·71-s − 0.702·73-s − 0.455·77-s + 1.46·79-s + 0.109·83-s − 1.58·89-s + 1.88·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36578304\)    =    \(2^{10} \cdot 3^{6} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.196958552$
$L(\frac12)$  $\approx$  $1.196958552$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 9 T + 42 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 15 T + 110 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 19 T + 168 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 19 T + 186 T^{2} - 19 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 138 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 13 T + 196 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T + 162 T^{2} - p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.185037566124457454187965356609, −8.116449456765153032038800563412, −7.28335805555971787475607674676, −7.15250902348865712865562849079, −6.86179672765516655668586464395, −6.60745435867943226993054356340, −6.07471152038805856006886508985, −5.95884525200823944220628810270, −5.16660773076303965196930016671, −5.06619148999116622694740139628, −4.65241313871062041872323115592, −4.46123939242294234878767897192, −3.73985436468492827761787321342, −3.39174930978679927647151231476, −2.91645301135024965335950802141, −2.58925734690730868482804582221, −2.17154085802695561029573545373, −1.69378607093471906529247316726, −0.995088022773706210693827393999, −0.30485207844741623894514425027, 0.30485207844741623894514425027, 0.995088022773706210693827393999, 1.69378607093471906529247316726, 2.17154085802695561029573545373, 2.58925734690730868482804582221, 2.91645301135024965335950802141, 3.39174930978679927647151231476, 3.73985436468492827761787321342, 4.46123939242294234878767897192, 4.65241313871062041872323115592, 5.06619148999116622694740139628, 5.16660773076303965196930016671, 5.95884525200823944220628810270, 6.07471152038805856006886508985, 6.60745435867943226993054356340, 6.86179672765516655668586464395, 7.15250902348865712865562849079, 7.28335805555971787475607674676, 8.116449456765153032038800563412, 8.185037566124457454187965356609

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