Properties

Label 4-4235e2-1.1-c1e2-0-2
Degree $4$
Conductor $17935225$
Sign $1$
Analytic cond. $1143.56$
Root an. cond. $5.81520$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s − 2·5-s + 2·7-s − 4·9-s − 12-s − 7·13-s + 2·15-s − 3·16-s − 9·17-s + 2·19-s − 2·20-s − 2·21-s + 3·25-s + 6·27-s + 2·28-s + 5·29-s + 2·31-s − 4·35-s − 4·36-s − 10·37-s + 7·39-s + 4·41-s + 6·43-s + 8·45-s − 9·47-s + 3·48-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.755·7-s − 4/3·9-s − 0.288·12-s − 1.94·13-s + 0.516·15-s − 3/4·16-s − 2.18·17-s + 0.458·19-s − 0.447·20-s − 0.436·21-s + 3/5·25-s + 1.15·27-s + 0.377·28-s + 0.928·29-s + 0.359·31-s − 0.676·35-s − 2/3·36-s − 1.64·37-s + 1.12·39-s + 0.624·41-s + 0.914·43-s + 1.19·45-s − 1.31·47-s + 0.433·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17935225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17935225\)    =    \(5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1143.56\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4235} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 17935225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 53 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 14 T + 166 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 15 T + 201 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - T + 65 T^{2} - p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 15 T + 189 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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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

−8.092582659500586959166230802670, −7.86995788280951749911567349659, −7.46465613488651781385076081105, −7.13947161232887866348855133697, −6.71859290254297040118163411549, −6.43206868596616074682704105938, −6.17792246521172978695981051175, −5.55670280602890024036052318473, −4.97590174638285796663307971688, −4.95039088217076630484222308665, −4.55749428408365161686413029873, −4.29780562648481389884533158451, −3.49912845228479082262868943128, −3.07106680839664990760492274157, −2.68034246250065049868440454068, −2.08133672554044300140068799309, −2.04858530132355269273641636384, −0.920308906870214704398971553972, 0, 0, 0.920308906870214704398971553972, 2.04858530132355269273641636384, 2.08133672554044300140068799309, 2.68034246250065049868440454068, 3.07106680839664990760492274157, 3.49912845228479082262868943128, 4.29780562648481389884533158451, 4.55749428408365161686413029873, 4.95039088217076630484222308665, 4.97590174638285796663307971688, 5.55670280602890024036052318473, 6.17792246521172978695981051175, 6.43206868596616074682704105938, 6.71859290254297040118163411549, 7.13947161232887866348855133697, 7.46465613488651781385076081105, 7.86995788280951749911567349659, 8.092582659500586959166230802670

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