Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·7-s + 4·11-s − 6·17-s − 8·19-s − 8·23-s + 25-s + 8·31-s − 4·35-s − 6·37-s − 2·41-s − 10·43-s + 6·47-s + 3·49-s + 4·53-s + 8·55-s − 10·59-s + 12·61-s − 8·67-s − 8·71-s − 8·77-s − 2·79-s − 22·83-s − 12·85-s + 12·89-s − 16·95-s − 8·97-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s + 1.20·11-s − 1.45·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.43·31-s − 0.676·35-s − 0.986·37-s − 0.312·41-s − 1.52·43-s + 0.875·47-s + 3/7·49-s + 0.549·53-s + 1.07·55-s − 1.30·59-s + 1.53·61-s − 0.977·67-s − 0.949·71-s − 0.911·77-s − 0.225·79-s − 2.41·83-s − 1.30·85-s + 1.27·89-s − 1.64·95-s − 0.812·97-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  =  2
Selberg data  =  $(4,\ 36578304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 103 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 10 T + 111 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 151 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 22 T + 255 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 8 T + 10 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−7.81011326268533348352262089541, −7.65396321981258921492626199941, −6.87501472250387913088680690291, −6.72119574018084756204827585770, −6.41225230463595037920270164823, −6.38586902522893257380633183651, −5.79537085963356840706047224756, −5.59367111080478406704448823629, −4.97586812430340030043080397034, −4.52157169798786771320585054419, −4.17049172428244662100081706262, −3.99277943782982676958498045625, −3.50449785333652596375099924501, −2.91984845966081298353489610819, −2.29968863771069370327941708443, −2.29742230343259759493149632541, −1.58298195948073124474948445394, −1.26124185778747771562499140767, 0, 0, 1.26124185778747771562499140767, 1.58298195948073124474948445394, 2.29742230343259759493149632541, 2.29968863771069370327941708443, 2.91984845966081298353489610819, 3.50449785333652596375099924501, 3.99277943782982676958498045625, 4.17049172428244662100081706262, 4.52157169798786771320585054419, 4.97586812430340030043080397034, 5.59367111080478406704448823629, 5.79537085963356840706047224756, 6.38586902522893257380633183651, 6.41225230463595037920270164823, 6.72119574018084756204827585770, 6.87501472250387913088680690291, 7.65396321981258921492626199941, 7.81011326268533348352262089541

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