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 + 12·29-s − 8·31-s + 20·41-s + 8·44-s − 49-s + 8·59-s − 16·61-s − 64-s − 4·71-s + 12·76-s − 32·79-s − 4·89-s − 12·101-s + 36·109-s − 12·116-s + 26·121-s + 8·124-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 + 2.22·29-s − 1.43·31-s + 3.12·41-s + 1.20·44-s − 1/7·49-s + 1.04·59-s − 2.04·61-s − 1/8·64-s − 0.474·71-s + 1.37·76-s − 3.60·79-s − 0.423·89-s − 1.19·101-s + 3.44·109-s − 1.11·116-s + 2.36·121-s + 0.718·124-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  =  $(4,\ 9922500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.2548044428$
$L(\frac12)$  $\approx$  $0.2548044428$
$L(\frac{3}{2})$   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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−8.728574922293480237348553011838, −8.663187105128396370217008490890, −8.160618847450276972319110530546, −7.61865278886682822730325139366, −7.61621120401563747472151358691, −7.14694670240559627966143056293, −6.43161845270593378323672486691, −6.28220007064314980836071518760, −5.67488746212090373125868442909, −5.61545149263463596047728862679, −4.86627943848617486282116948084, −4.68397690188515935167409193995, −4.16594551363586349195500193966, −4.05239657424506183186052804338, −3.06729337701200944141762454417, −2.81251756406472609130400871824, −2.39856718184895471042088853610, −1.95996452690452873532812760320, −1.07983564110204276269333971784, −0.16961349179264375423260645715, 0.16961349179264375423260645715, 1.07983564110204276269333971784, 1.95996452690452873532812760320, 2.39856718184895471042088853610, 2.81251756406472609130400871824, 3.06729337701200944141762454417, 4.05239657424506183186052804338, 4.16594551363586349195500193966, 4.68397690188515935167409193995, 4.86627943848617486282116948084, 5.61545149263463596047728862679, 5.67488746212090373125868442909, 6.28220007064314980836071518760, 6.43161845270593378323672486691, 7.14694670240559627966143056293, 7.61621120401563747472151358691, 7.61865278886682822730325139366, 8.160618847450276972319110530546, 8.663187105128396370217008490890, 8.728574922293480237348553011838

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