Properties

Label 4-630e2-1.1-c1e2-0-18
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
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 + 16·19-s + 4·20-s + 11·25-s − 16·29-s + 8·31-s + 24·41-s − 49-s − 16·59-s − 12·61-s − 64-s − 16·76-s − 16·79-s − 4·80-s + 8·89-s − 64·95-s − 11·100-s + 24·101-s − 4·109-s + 16·116-s − 22·121-s − 8·124-s − 24·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s + 1/4·16-s + 3.67·19-s + 0.894·20-s + 11/5·25-s − 2.97·29-s + 1.43·31-s + 3.74·41-s − 1/7·49-s − 2.08·59-s − 1.53·61-s − 1/8·64-s − 1.83·76-s − 1.80·79-s − 0.447·80-s + 0.847·89-s − 6.56·95-s − 1.09·100-s + 2.38·101-s − 0.383·109-s + 1.48·116-s − 2·121-s − 0.718·124-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.069783821\)
\(L(\frac12)\) \(\approx\) \(1.069783821\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−10.91946341175581805790662520964, −10.49839725824576295199357472012, −9.750325669363806175511469432261, −9.375143355219890940722532391734, −9.231004043744810732276820062365, −8.710471852531880161530895307943, −7.86094990335840108523802619017, −7.70978173658112462198926068231, −7.40815005074694681976714668290, −7.29784820732422506511104865715, −6.16337119095468751719743091885, −5.86477934218199323320095027219, −5.16243260990912172191610139147, −4.83849137185205450889737786480, −4.09237732048846523750102007496, −3.85889533379040075872733995812, −3.06440711733884952843546864526, −2.91353021490250363503403738067, −1.40245221211779527951862810289, −0.63892303427725341813276550352, 0.63892303427725341813276550352, 1.40245221211779527951862810289, 2.91353021490250363503403738067, 3.06440711733884952843546864526, 3.85889533379040075872733995812, 4.09237732048846523750102007496, 4.83849137185205450889737786480, 5.16243260990912172191610139147, 5.86477934218199323320095027219, 6.16337119095468751719743091885, 7.29784820732422506511104865715, 7.40815005074694681976714668290, 7.70978173658112462198926068231, 7.86094990335840108523802619017, 8.710471852531880161530895307943, 9.231004043744810732276820062365, 9.375143355219890940722532391734, 9.750325669363806175511469432261, 10.49839725824576295199357472012, 10.91946341175581805790662520964

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