Properties

Label 4-1890e2-1.1-c1e2-0-23
Degree $4$
Conductor $3572100$
Sign $1$
Analytic cond. $227.760$
Root an. cond. $3.88480$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·5-s + 16-s − 8·19-s − 4·20-s + 11·25-s − 14·29-s + 2·31-s − 18·41-s − 49-s − 2·59-s + 6·61-s − 64-s − 6·71-s + 8·76-s + 32·79-s + 4·80-s − 20·89-s − 32·95-s − 11·100-s − 12·101-s + 32·109-s + 14·116-s − 22·121-s − 2·124-s + 24·125-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s + 1/4·16-s − 1.83·19-s − 0.894·20-s + 11/5·25-s − 2.59·29-s + 0.359·31-s − 2.81·41-s − 1/7·49-s − 0.260·59-s + 0.768·61-s − 1/8·64-s − 0.712·71-s + 0.917·76-s + 3.60·79-s + 0.447·80-s − 2.11·89-s − 3.28·95-s − 1.09·100-s − 1.19·101-s + 3.06·109-s + 1.29·116-s − 2·121-s − 0.179·124-s + 2.14·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3572100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3572100\)    =    \(2^{2} \cdot 3^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(227.760\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3572100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.018070212\)
\(L(\frac12)\) \(\approx\) \(2.018070212\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.470091773546405513123459425298, −9.149770123829168611879837226981, −8.694362821051594465767223036792, −8.284461468290240471856276836062, −8.093620136275700955652672827060, −7.38594591260685691947817269868, −6.79714984840198387560668406763, −6.72044235117819441967683221610, −6.21313126124882475859604742612, −5.75358463465654804451540163963, −5.32500589981550152037896970706, −5.24436121810505998525943837523, −4.38640417588946615332972826515, −4.28792558143184170364597357479, −3.38247165199816890224337153379, −3.20396870026367943084872760203, −2.23794449331697112902346353363, −1.94400730203552119912087481795, −1.62738591048487181518791342440, −0.49759246455891484856990489852, 0.49759246455891484856990489852, 1.62738591048487181518791342440, 1.94400730203552119912087481795, 2.23794449331697112902346353363, 3.20396870026367943084872760203, 3.38247165199816890224337153379, 4.28792558143184170364597357479, 4.38640417588946615332972826515, 5.24436121810505998525943837523, 5.32500589981550152037896970706, 5.75358463465654804451540163963, 6.21313126124882475859604742612, 6.72044235117819441967683221610, 6.79714984840198387560668406763, 7.38594591260685691947817269868, 8.093620136275700955652672827060, 8.284461468290240471856276836062, 8.694362821051594465767223036792, 9.149770123829168611879837226981, 9.470091773546405513123459425298

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