Properties

Label 4-35e4-1.1-c0e2-0-0
Degree $4$
Conductor $1500625$
Sign $1$
Analytic cond. $0.373754$
Root an. cond. $0.781891$
Motivic weight $0$
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  − 2-s + 4-s − 2·8-s − 9-s + 11-s + 2·16-s + 18-s − 22-s − 23-s − 2·29-s − 2·32-s − 36-s − 37-s + 2·43-s + 44-s + 46-s + 2·53-s + 2·58-s + 3·64-s − 67-s − 2·71-s + 2·72-s + 74-s + 79-s − 2·86-s − 2·88-s − 92-s + ⋯
L(s)  = 1  − 2-s + 4-s − 2·8-s − 9-s + 11-s + 2·16-s + 18-s − 22-s − 23-s − 2·29-s − 2·32-s − 36-s − 37-s + 2·43-s + 44-s + 46-s + 2·53-s + 2·58-s + 3·64-s − 67-s − 2·71-s + 2·72-s + 74-s + 79-s − 2·86-s − 2·88-s − 92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1500625\)    =    \(5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.373754\)
Root analytic conductor: \(0.781891\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1500625,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4857522993\)
\(L(\frac12)\) \(\approx\) \(0.4857522993\)
\(L(1)\) 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)$
bad5 \( 1 \)
7 \( 1 \)
good2$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 - T + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_2$ \( ( 1 + T + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.03313300817045812074929488173, −9.578040823003165920550467263826, −9.229758580470922955291942117830, −8.924255916202679686666584223595, −8.576136475883728372990583116844, −8.372531608678366215526791706188, −7.59842176902218999053938089307, −7.30725607402941585738144896994, −7.04006346212564303124405887976, −6.19099761555383735231315200215, −6.13763063646913018186982607990, −5.57428619416559246494646842947, −5.49525113082812853994718989058, −4.37372423176520755756869262364, −3.98645908157874496953427827379, −3.20266447089468753644929897930, −3.15985079605847204754569431114, −2.10925468469655311819240858735, −1.95256991266311545215048999760, −0.69949591763161655647229985242, 0.69949591763161655647229985242, 1.95256991266311545215048999760, 2.10925468469655311819240858735, 3.15985079605847204754569431114, 3.20266447089468753644929897930, 3.98645908157874496953427827379, 4.37372423176520755756869262364, 5.49525113082812853994718989058, 5.57428619416559246494646842947, 6.13763063646913018186982607990, 6.19099761555383735231315200215, 7.04006346212564303124405887976, 7.30725607402941585738144896994, 7.59842176902218999053938089307, 8.372531608678366215526791706188, 8.576136475883728372990583116844, 8.924255916202679686666584223595, 9.229758580470922955291942117830, 9.578040823003165920550467263826, 10.03313300817045812074929488173

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