Properties

Label 4-585e2-1.1-c1e2-0-2
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  + 2·2-s − 4-s + 2·5-s − 8·8-s + 4·10-s + 6·13-s − 7·16-s − 2·20-s − 25-s + 12·26-s − 12·29-s + 14·32-s + 12·37-s − 16·40-s + 16·47-s − 14·49-s − 2·50-s − 6·52-s − 24·58-s + 12·61-s + 35·64-s + 12·65-s + 24·67-s − 12·73-s + 24·74-s − 14·80-s + 8·83-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s + 0.894·5-s − 2.82·8-s + 1.26·10-s + 1.66·13-s − 7/4·16-s − 0.447·20-s − 1/5·25-s + 2.35·26-s − 2.22·29-s + 2.47·32-s + 1.97·37-s − 2.52·40-s + 2.33·47-s − 2·49-s − 0.282·50-s − 0.832·52-s − 3.15·58-s + 1.53·61-s + 35/8·64-s + 1.48·65-s + 2.93·67-s − 1.40·73-s + 2.78·74-s − 1.56·80-s + 0.878·83-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: induced by $\chi_{585} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.858907793\)
\(L(\frac12)\) \(\approx\) \(2.858907793\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−11.06325968634500378071062598839, −10.56748351633643151661211230719, −9.826968328721183710495174574346, −9.463905561104978022490565122142, −9.396465806080167156949871379253, −8.820292448823425517429833150819, −8.211690105870193745802521238935, −8.108270293358545525105978610700, −7.19007111162300088656871640068, −6.53908841650111127574115009615, −6.07866350853521321111553730291, −5.68693915005834351095918519264, −5.53392881829809494019240713335, −4.92161691746958411873039800166, −4.11912991484653771899845166361, −3.99854341562248601697793958572, −3.44721758731866515998227472013, −2.76524914825219347389641769751, −1.95427287501776239266919869683, −0.802392914566662689323310059992, 0.802392914566662689323310059992, 1.95427287501776239266919869683, 2.76524914825219347389641769751, 3.44721758731866515998227472013, 3.99854341562248601697793958572, 4.11912991484653771899845166361, 4.92161691746958411873039800166, 5.53392881829809494019240713335, 5.68693915005834351095918519264, 6.07866350853521321111553730291, 6.53908841650111127574115009615, 7.19007111162300088656871640068, 8.108270293358545525105978610700, 8.211690105870193745802521238935, 8.820292448823425517429833150819, 9.396465806080167156949871379253, 9.463905561104978022490565122142, 9.826968328721183710495174574346, 10.56748351633643151661211230719, 11.06325968634500378071062598839

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