Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 23^{2} \cdot 29^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 3·4-s + 8·5-s + 4·6-s + 4·7-s + 4·8-s + 3·9-s + 16·10-s + 6·12-s − 4·13-s + 8·14-s + 16·15-s + 5·16-s + 6·18-s − 4·19-s + 24·20-s + 8·21-s − 2·23-s + 8·24-s + 38·25-s − 8·26-s + 4·27-s + 12·28-s + 2·29-s + 32·30-s − 12·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 3.57·5-s + 1.63·6-s + 1.51·7-s + 1.41·8-s + 9-s + 5.05·10-s + 1.73·12-s − 1.10·13-s + 2.13·14-s + 4.13·15-s + 5/4·16-s + 1.41·18-s − 0.917·19-s + 5.36·20-s + 1.74·21-s − 0.417·23-s + 1.63·24-s + 38/5·25-s − 1.56·26-s + 0.769·27-s + 2.26·28-s + 0.371·29-s + 5.84·30-s − 2.15·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16016004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16016004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16016004\)    =    \(2^{2} \cdot 3^{2} \cdot 23^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4002} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 16016004,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(32.88460807\)
\(L(\frac12)\)  \(\approx\)  \(32.88460807\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;23,\;29\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 8 T + 74 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 20 T + 180 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 8 T + 176 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T - 96 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−8.632939848226437503780191135231, −8.403090421328483914470569760146, −7.68474069536071076522704141009, −7.55197030579555448635151370596, −6.88355179466220170772472571999, −6.86220626190112201930814571270, −6.20688874064931707119253342254, −5.93677371692860366359229954731, −5.57786271831326875210536455653, −5.30488739505490042481655884485, −4.89548307172694561832298614343, −4.69873326243756692345218101486, −4.12525737397461835889529624963, −3.67721095474791598088477050452, −2.86364388705708469923587278650, −2.63328223493268613801757118097, −2.21983711838196141782436989069, −2.00573873507345237659337128524, −1.57893789606784592363738379877, −1.29017147924949527991854214699, 1.29017147924949527991854214699, 1.57893789606784592363738379877, 2.00573873507345237659337128524, 2.21983711838196141782436989069, 2.63328223493268613801757118097, 2.86364388705708469923587278650, 3.67721095474791598088477050452, 4.12525737397461835889529624963, 4.69873326243756692345218101486, 4.89548307172694561832298614343, 5.30488739505490042481655884485, 5.57786271831326875210536455653, 5.93677371692860366359229954731, 6.20688874064931707119253342254, 6.86220626190112201930814571270, 6.88355179466220170772472571999, 7.55197030579555448635151370596, 7.68474069536071076522704141009, 8.403090421328483914470569760146, 8.632939848226437503780191135231

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