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  − 4-s − 8·11-s + 16-s + 12·19-s + 6·29-s − 14·31-s + 2·41-s + 8·44-s − 49-s − 22·59-s + 26·61-s − 64-s − 16·71-s − 12·76-s − 8·79-s − 28·89-s + 12·101-s + 12·109-s − 6·116-s + 26·121-s + 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 2.41·11-s + 1/4·16-s + 2.75·19-s + 1.11·29-s − 2.51·31-s + 0.312·41-s + 1.20·44-s − 1/7·49-s − 2.86·59-s + 3.32·61-s − 1/8·64-s − 1.89·71-s − 1.37·76-s − 0.900·79-s − 2.96·89-s + 1.19·101-s + 1.14·109-s − 0.557·116-s + 2.36·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.8713335795$
$L(\frac12)$  $\approx$  $0.8713335795$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.971814274350725086893985766304, −8.520889850402485317293226546352, −7.924726274951671188813213117863, −7.68838988493980913782514098621, −7.64528950345514388191840347560, −6.98139296856741614280793876088, −6.91511882506102448207047928237, −5.93877688352697497386510716128, −5.74843584378754269945124412371, −5.31930590483731274998713153941, −5.21222496618157481439847291050, −4.73836620825753519083624131748, −4.28051341713366658861968841436, −3.62037561762735587466915906076, −3.27121004810812307062897605064, −2.80854997288585648059657907601, −2.54719009480443525736089113299, −1.71255675964978544711776484781, −1.15903430291555601009020863450, −0.31349598183089335991074913843, 0.31349598183089335991074913843, 1.15903430291555601009020863450, 1.71255675964978544711776484781, 2.54719009480443525736089113299, 2.80854997288585648059657907601, 3.27121004810812307062897605064, 3.62037561762735587466915906076, 4.28051341713366658861968841436, 4.73836620825753519083624131748, 5.21222496618157481439847291050, 5.31930590483731274998713153941, 5.74843584378754269945124412371, 5.93877688352697497386510716128, 6.91511882506102448207047928237, 6.98139296856741614280793876088, 7.64528950345514388191840347560, 7.68838988493980913782514098621, 7.924726274951671188813213117863, 8.520889850402485317293226546352, 8.971814274350725086893985766304

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