Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 41^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 2·9-s + 5·16-s + 4·18-s − 10·25-s − 8·31-s − 6·32-s − 6·36-s + 4·37-s + 6·41-s + 16·43-s + 49-s + 20·50-s − 12·59-s + 16·61-s + 16·62-s + 7·64-s + 8·72-s + 4·73-s − 8·74-s − 5·81-s − 12·82-s − 12·83-s − 32·86-s − 2·98-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2/3·9-s + 5/4·16-s + 0.942·18-s − 2·25-s − 1.43·31-s − 1.06·32-s − 36-s + 0.657·37-s + 0.937·41-s + 2.43·43-s + 1/7·49-s + 2.82·50-s − 1.56·59-s + 2.04·61-s + 2.03·62-s + 7/8·64-s + 0.942·72-s + 0.468·73-s − 0.929·74-s − 5/9·81-s − 1.32·82-s − 1.31·83-s − 3.45·86-s − 0.202·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(329476\)    =    \(2^{2} \cdot 7^{2} \cdot 41^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{329476} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 329476,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;41\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
41$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.576574175927212899111454819532, −8.131298345473117072946549713145, −7.57571100088867902110310233811, −7.46852791716783421092039703831, −6.86761839836579218858889642687, −6.14364779323247904643646496119, −5.76672260141591023723183382906, −5.57928681742950427486583645839, −4.55033994430147192557970938084, −3.92908850742833290739203075499, −3.35438962358155565091884257335, −2.46869134005901036292907277411, −2.14361086660228166648215843038, −1.11551143424508389710750581105, 0, 1.11551143424508389710750581105, 2.14361086660228166648215843038, 2.46869134005901036292907277411, 3.35438962358155565091884257335, 3.92908850742833290739203075499, 4.55033994430147192557970938084, 5.57928681742950427486583645839, 5.76672260141591023723183382906, 6.14364779323247904643646496119, 6.86761839836579218858889642687, 7.46852791716783421092039703831, 7.57571100088867902110310233811, 8.131298345473117072946549713145, 8.576574175927212899111454819532

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