Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{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 + 3·4-s + 2·7-s − 4·8-s − 4·13-s − 4·14-s + 5·16-s + 4·17-s + 8·19-s − 4·23-s + 8·26-s + 6·28-s − 4·29-s + 8·31-s − 6·32-s − 8·34-s − 4·37-s − 16·38-s + 12·41-s − 8·43-s + 8·46-s + 8·47-s + 3·49-s − 12·52-s − 12·53-s − 8·56-s + 8·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 1.10·13-s − 1.06·14-s + 5/4·16-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 1.56·26-s + 1.13·28-s − 0.742·29-s + 1.43·31-s − 1.06·32-s − 1.37·34-s − 0.657·37-s − 2.59·38-s + 1.87·41-s − 1.21·43-s + 1.17·46-s + 1.16·47-s + 3/7·49-s − 1.66·52-s − 1.64·53-s − 1.06·56-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(9922500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{3150} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.763607605$
$L(\frac12)$  $\approx$  $1.763607605$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, 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 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 128 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 12 T + 152 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−8.748720080240834891533252527474, −8.482695314716223891060835874297, −7.954365024723372766470644006249, −7.945031765864917522280222777612, −7.50278031899873678476055597616, −7.25302146126156984086378209602, −6.65841241549936900798256318312, −6.55043219412069728060396235264, −5.73838245348089955810376592647, −5.55099238941352453213462199316, −5.10518644789293325280282707379, −4.85373147924713434409810021807, −3.93562728597974423781216070489, −3.83692107553771915336431592214, −2.91011075334615027338934830526, −2.86580594149057280301441010398, −1.98663644315735309451399381304, −1.85788147732680409848894070100, −0.847387495208084136665961882814, −0.72590807720796895153372359854, 0.72590807720796895153372359854, 0.847387495208084136665961882814, 1.85788147732680409848894070100, 1.98663644315735309451399381304, 2.86580594149057280301441010398, 2.91011075334615027338934830526, 3.83692107553771915336431592214, 3.93562728597974423781216070489, 4.85373147924713434409810021807, 5.10518644789293325280282707379, 5.55099238941352453213462199316, 5.73838245348089955810376592647, 6.55043219412069728060396235264, 6.65841241549936900798256318312, 7.25302146126156984086378209602, 7.50278031899873678476055597616, 7.945031765864917522280222777612, 7.954365024723372766470644006249, 8.482695314716223891060835874297, 8.748720080240834891533252527474

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