Properties

Label 4-585e2-1.1-c1e2-0-7
Degree $4$
Conductor $342225$
Sign $1$
Analytic cond. $21.8205$
Root an. cond. $2.16130$
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  + 3·4-s + 6·13-s + 5·16-s − 8·17-s − 16·23-s − 25-s + 16·29-s + 16·43-s − 2·49-s + 18·52-s + 24·53-s − 4·61-s + 3·64-s − 24·68-s − 16·79-s − 48·92-s − 3·100-s + 32·101-s + 8·103-s + 16·107-s + 8·113-s + 48·116-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.66·13-s + 5/4·16-s − 1.94·17-s − 3.33·23-s − 1/5·25-s + 2.97·29-s + 2.43·43-s − 2/7·49-s + 2.49·52-s + 3.29·53-s − 0.512·61-s + 3/8·64-s − 2.91·68-s − 1.80·79-s − 5.00·92-s − 0.299·100-s + 3.18·101-s + 0.788·103-s + 1.54·107-s + 0.752·113-s + 4.45·116-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(342225\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.8205\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.743332040\)
\(L(\frac12)\) \(\approx\) \(2.743332040\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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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

−11.07349432980446025995960795723, −10.38559214178587156418313538205, −10.20997781130083069713440051899, −9.953440349825541674078577629513, −8.859929457988981522701384298496, −8.696605152572893461376411603374, −8.424628054204083693014282226639, −7.70627867723409188554045580641, −7.35957728137547715617732859206, −6.80451348856461094121887567165, −6.24939274736086820220532197566, −5.97541394162336676655580748258, −5.94857462281854737758207200707, −4.72929811400158416163632280812, −4.19401162936134591544195603344, −3.86655853655184341389840404198, −3.00402961081401192024180662387, −2.17986464670808874121040457053, −2.14955067662077876701430927224, −0.950734341939368621729886097054, 0.950734341939368621729886097054, 2.14955067662077876701430927224, 2.17986464670808874121040457053, 3.00402961081401192024180662387, 3.86655853655184341389840404198, 4.19401162936134591544195603344, 4.72929811400158416163632280812, 5.94857462281854737758207200707, 5.97541394162336676655580748258, 6.24939274736086820220532197566, 6.80451348856461094121887567165, 7.35957728137547715617732859206, 7.70627867723409188554045580641, 8.424628054204083693014282226639, 8.696605152572893461376411603374, 8.859929457988981522701384298496, 9.953440349825541674078577629513, 10.20997781130083069713440051899, 10.38559214178587156418313538205, 11.07349432980446025995960795723

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