Properties

Label 4-1305e2-1.1-c1e2-0-4
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $108.586$
Root an. cond. $3.22807$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s − 4·5-s + 8·8-s − 16·10-s − 4·16-s + 12·17-s − 32·20-s + 11·25-s + 4·29-s − 32·32-s + 48·34-s − 2·37-s − 32·40-s − 2·43-s − 16·47-s − 2·49-s + 44·50-s + 16·58-s − 16·59-s − 64·64-s + 96·68-s − 4·71-s + 30·73-s − 8·74-s + 16·80-s − 48·85-s + ⋯
L(s)  = 1  + 2.82·2-s + 4·4-s − 1.78·5-s + 2.82·8-s − 5.05·10-s − 16-s + 2.91·17-s − 7.15·20-s + 11/5·25-s + 0.742·29-s − 5.65·32-s + 8.23·34-s − 0.328·37-s − 5.05·40-s − 0.304·43-s − 2.33·47-s − 2/7·49-s + 6.22·50-s + 2.10·58-s − 2.08·59-s − 8·64-s + 11.6·68-s − 0.474·71-s + 3.51·73-s − 0.929·74-s + 1.78·80-s − 5.20·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(108.586\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.019573497\)
\(L(\frac12)\) \(\approx\) \(6.019573497\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
29$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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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

−9.738429152327209098155837035283, −9.705241165562487214321846287697, −8.950893201578666372078248978392, −8.302843257897433570293909223490, −8.207370617276444456773597116117, −7.60973984109287156553467338298, −7.27300529849285495151592315098, −6.83579944325227858511132406863, −6.14885502234064744496038314249, −6.13340633299569553057515089107, −5.40613569696574780957360186736, −4.95488288955330625006751518500, −4.80012059859727431091523076309, −4.37797694144427953366951522619, −3.59254007917073859749384693798, −3.47663691456538662284798495539, −3.28478753548417363019912845808, −2.76150253355720034799778383644, −1.77088019764797899931344369182, −0.64223607711012492924852643119, 0.64223607711012492924852643119, 1.77088019764797899931344369182, 2.76150253355720034799778383644, 3.28478753548417363019912845808, 3.47663691456538662284798495539, 3.59254007917073859749384693798, 4.37797694144427953366951522619, 4.80012059859727431091523076309, 4.95488288955330625006751518500, 5.40613569696574780957360186736, 6.13340633299569553057515089107, 6.14885502234064744496038314249, 6.83579944325227858511132406863, 7.27300529849285495151592315098, 7.60973984109287156553467338298, 8.207370617276444456773597116117, 8.302843257897433570293909223490, 8.950893201578666372078248978392, 9.705241165562487214321846287697, 9.738429152327209098155837035283

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