Properties

Label 4-1305e2-1.1-c1e2-0-11
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $108.586$
Root an. cond. $3.22807$
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  − 4·2-s + 8·4-s − 4·5-s − 8·8-s + 16·10-s − 4·16-s − 12·17-s − 32·20-s + 11·25-s + 4·29-s + 32·32-s + 48·34-s + 2·37-s + 32·40-s + 2·43-s + 16·47-s − 2·49-s − 44·50-s − 16·58-s − 16·59-s − 64·64-s − 96·68-s − 4·71-s − 30·73-s − 8·74-s + 16·80-s + 48·85-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 1.78·5-s − 2.82·8-s + 5.05·10-s − 16-s − 2.91·17-s − 7.15·20-s + 11/5·25-s + 0.742·29-s + 5.65·32-s + 8.23·34-s + 0.328·37-s + 5.05·40-s + 0.304·43-s + 2.33·47-s − 2/7·49-s − 6.22·50-s − 2.10·58-s − 2.08·59-s − 8·64-s − 11.6·68-s − 0.474·71-s − 3.51·73-s − 0.929·74-s + 1.78·80-s + 5.20·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(108.586\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1703025,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
29$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
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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

−9.258660502784135208864693518049, −8.869196131608214805979323256530, −8.669966985557616125093264499971, −8.485880781769781364752005316104, −7.981911337327010903605946279591, −7.50129724541120000400336987757, −7.19653390109522233264405847812, −7.15300607786970201737044646277, −6.30165696495877185097365444372, −6.28689548255061591531898551684, −5.05519616446583229455787309199, −4.55906861670702344068536799101, −4.24192291234038408949871749819, −3.96689791974425177509330315264, −2.69098275202023595336610712018, −2.64506926223328858397976586206, −1.67219411835434745405184759008, −1.04089799470164135561260760267, 0, 0, 1.04089799470164135561260760267, 1.67219411835434745405184759008, 2.64506926223328858397976586206, 2.69098275202023595336610712018, 3.96689791974425177509330315264, 4.24192291234038408949871749819, 4.55906861670702344068536799101, 5.05519616446583229455787309199, 6.28689548255061591531898551684, 6.30165696495877185097365444372, 7.15300607786970201737044646277, 7.19653390109522233264405847812, 7.50129724541120000400336987757, 7.981911337327010903605946279591, 8.485880781769781364752005316104, 8.669966985557616125093264499971, 8.869196131608214805979323256530, 9.258660502784135208864693518049

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