Properties

Label 4-9295e2-1.1-c1e2-0-0
Degree $4$
Conductor $86397025$
Sign $1$
Analytic cond. $5508.74$
Root an. cond. $8.61515$
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 + 4·7-s + 2·9-s − 4·10-s − 2·11-s − 8·14-s + 16-s + 8·17-s − 4·18-s + 2·20-s + 4·22-s + 3·25-s + 4·28-s + 4·29-s + 2·32-s − 16·34-s + 8·35-s + 2·36-s + 4·37-s − 12·41-s − 12·43-s − 2·44-s + 4·45-s − 2·49-s − 6·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s + 2/3·9-s − 1.26·10-s − 0.603·11-s − 2.13·14-s + 1/4·16-s + 1.94·17-s − 0.942·18-s + 0.447·20-s + 0.852·22-s + 3/5·25-s + 0.755·28-s + 0.742·29-s + 0.353·32-s − 2.74·34-s + 1.35·35-s + 1/3·36-s + 0.657·37-s − 1.87·41-s − 1.82·43-s − 0.301·44-s + 0.596·45-s − 2/7·49-s − 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86397025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86397025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86397025\)    =    \(5^{2} \cdot 11^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(5508.74\)
Root analytic conductor: \(8.61515\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9295} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86397025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.294613095\)
\(L(\frac12)\) \(\approx\) \(2.294613095\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.072030087044854451712609832327, −7.62926153762886216492375361216, −7.30006999382417069963732032153, −7.17228407899312398049856832382, −6.45363616445473831613587675061, −6.30475046427185450448657535679, −5.92897779151536536843196476874, −5.22545348697061647241077523690, −5.19377388367535862884396994286, −5.01938373227901756208932977384, −4.53401267986844259817562140131, −4.05630227843943496546017351971, −3.45003488722167814650762971944, −3.22219061005423894264054353909, −2.72503746672991041715810033517, −1.96980343936148011121971291438, −1.93781508675787687574555098567, −1.35502173047735544451234590375, −0.952888085064269769201358610919, −0.53629556289382694327575866400, 0.53629556289382694327575866400, 0.952888085064269769201358610919, 1.35502173047735544451234590375, 1.93781508675787687574555098567, 1.96980343936148011121971291438, 2.72503746672991041715810033517, 3.22219061005423894264054353909, 3.45003488722167814650762971944, 4.05630227843943496546017351971, 4.53401267986844259817562140131, 5.01938373227901756208932977384, 5.19377388367535862884396994286, 5.22545348697061647241077523690, 5.92897779151536536843196476874, 6.30475046427185450448657535679, 6.45363616445473831613587675061, 7.17228407899312398049856832382, 7.30006999382417069963732032153, 7.62926153762886216492375361216, 8.072030087044854451712609832327

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