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 + 6·5-s + 4·6-s + 6·7-s + 4·8-s + 3·9-s + 12·10-s − 2·11-s + 6·12-s + 6·13-s + 12·14-s + 12·15-s + 5·16-s − 6·17-s + 6·18-s − 2·19-s + 18·20-s + 12·21-s − 4·22-s + 2·23-s + 8·24-s + 17·25-s + 12·26-s + 4·27-s + 18·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 2.68·5-s + 1.63·6-s + 2.26·7-s + 1.41·8-s + 9-s + 3.79·10-s − 0.603·11-s + 1.73·12-s + 1.66·13-s + 3.20·14-s + 3.09·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s − 0.458·19-s + 4.02·20-s + 2.61·21-s − 0.852·22-s + 0.417·23-s + 1.63·24-s + 17/5·25-s + 2.35·26-s + 0.769·27-s + 3.40·28-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  =  $(4,\ 16016004,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $32.13573044$
$L(\frac12)$  $\approx$  $32.13573044$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 155 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 36 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 218 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 170 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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

−8.605604054040269641245109358858, −8.521617222218064874829368939853, −7.79374147666536197674691367321, −7.56621916691075994063872995211, −7.03410575796934842151850213726, −6.70477771501253320983184293126, −6.24342437650037440482466507213, −5.98119933393527183218386044972, −5.48031554669020915385769218925, −5.38587185575190134227132702737, −4.75908611811333670646025170905, −4.70908447365688150814569212132, −3.98846431518905623848836320299, −3.84495065956566398078168293970, −2.98182890718561413265398533638, −2.66309208802909123954997922720, −2.24520048130962089081873197264, −1.75647714637458985883726461871, −1.62580224380410981962656552652, −1.33263568038846512018648282754, 1.33263568038846512018648282754, 1.62580224380410981962656552652, 1.75647714637458985883726461871, 2.24520048130962089081873197264, 2.66309208802909123954997922720, 2.98182890718561413265398533638, 3.84495065956566398078168293970, 3.98846431518905623848836320299, 4.70908447365688150814569212132, 4.75908611811333670646025170905, 5.38587185575190134227132702737, 5.48031554669020915385769218925, 5.98119933393527183218386044972, 6.24342437650037440482466507213, 6.70477771501253320983184293126, 7.03410575796934842151850213726, 7.56621916691075994063872995211, 7.79374147666536197674691367321, 8.521617222218064874829368939853, 8.605604054040269641245109358858

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