Properties

Label 4-2898e2-1.1-c1e2-0-8
Degree $4$
Conductor $8398404$
Sign $1$
Analytic cond. $535.489$
Root an. cond. $4.81047$
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  + 2·2-s + 3·4-s − 5-s − 2·7-s + 4·8-s − 2·10-s − 2·11-s − 3·13-s − 4·14-s + 5·16-s − 6·17-s − 2·19-s − 3·20-s − 4·22-s − 2·23-s − 5·25-s − 6·26-s − 6·28-s − 11·29-s − 2·31-s + 6·32-s − 12·34-s + 2·35-s + 37-s − 4·38-s − 4·40-s − 3·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.447·5-s − 0.755·7-s + 1.41·8-s − 0.632·10-s − 0.603·11-s − 0.832·13-s − 1.06·14-s + 5/4·16-s − 1.45·17-s − 0.458·19-s − 0.670·20-s − 0.852·22-s − 0.417·23-s − 25-s − 1.17·26-s − 1.13·28-s − 2.04·29-s − 0.359·31-s + 1.06·32-s − 2.05·34-s + 0.338·35-s + 0.164·37-s − 0.648·38-s − 0.632·40-s − 0.468·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8398404\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(535.489\)
Root analytic conductor: \(4.81047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2898} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8398404,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - T + 70 T^{2} - p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 29 T + 400 T^{2} + 29 p T^{3} + p^{2} T^{4} \)
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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

−8.328872305181468231792344693402, −8.137428607884339326372240436451, −7.64212240913575077579625388714, −7.45061820328268051734395708104, −6.84437884226441358308352963666, −6.70499397192025405259067539151, −6.18020019111576913208434725983, −5.96117013321385477602204595340, −5.29672860862609689290622670443, −5.16837199052561964432297922465, −4.62823137949894990766042509918, −4.19864030745313188478856111238, −3.82376625783524314410067799514, −3.53861950401774083328688677691, −2.88593429190987833786608650985, −2.55422667344097801323175761226, −1.96806931661533329899196232018, −1.63328528299370640231478682293, 0, 0, 1.63328528299370640231478682293, 1.96806931661533329899196232018, 2.55422667344097801323175761226, 2.88593429190987833786608650985, 3.53861950401774083328688677691, 3.82376625783524314410067799514, 4.19864030745313188478856111238, 4.62823137949894990766042509918, 5.16837199052561964432297922465, 5.29672860862609689290622670443, 5.96117013321385477602204595340, 6.18020019111576913208434725983, 6.70499397192025405259067539151, 6.84437884226441358308352963666, 7.45061820328268051734395708104, 7.64212240913575077579625388714, 8.137428607884339326372240436451, 8.328872305181468231792344693402

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