Properties

Label 4-1305e2-1.1-c1e2-0-0
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4-s + 2·5-s + 8·8-s − 4·10-s − 7·16-s − 12·17-s − 2·20-s − 25-s + 10·29-s − 14·32-s + 24·34-s + 4·37-s + 16·40-s − 8·43-s − 16·47-s + 10·49-s + 2·50-s − 20·58-s + 8·59-s + 35·64-s + 12·68-s − 16·71-s + 12·73-s − 8·74-s − 14·80-s − 24·85-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 0.894·5-s + 2.82·8-s − 1.26·10-s − 7/4·16-s − 2.91·17-s − 0.447·20-s − 1/5·25-s + 1.85·29-s − 2.47·32-s + 4.11·34-s + 0.657·37-s + 2.52·40-s − 1.21·43-s − 2.33·47-s + 10/7·49-s + 0.282·50-s − 2.62·58-s + 1.04·59-s + 35/8·64-s + 1.45·68-s − 1.89·71-s + 1.40·73-s − 0.929·74-s − 1.56·80-s − 2.60·85-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: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4488881226\)
\(L(\frac12)\) \(\approx\) \(0.4488881226\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
29$C_2$ \( 1 - 10 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−9.580522065062501171198349789066, −9.535540457390030055217666489754, −8.967798053409226243916777565384, −8.646583260804489178670566003207, −8.592230161283651090333120470135, −7.947221900702744398451124637937, −7.70736888774135246414550699110, −6.99088385583546517006611864781, −6.51210341109520376412783574737, −6.50730046317284448303947966062, −5.64925504501741451087805470001, −5.09511235598637249678840819526, −4.74606881653321709175340423170, −4.37150630020673572115594126454, −3.99533947688981946669488408711, −3.17056769883972852879849106608, −2.30739697223120162518397789661, −1.96548426644944456680066117736, −1.25225258836028428660641769657, −0.39321974284566799567496492888, 0.39321974284566799567496492888, 1.25225258836028428660641769657, 1.96548426644944456680066117736, 2.30739697223120162518397789661, 3.17056769883972852879849106608, 3.99533947688981946669488408711, 4.37150630020673572115594126454, 4.74606881653321709175340423170, 5.09511235598637249678840819526, 5.64925504501741451087805470001, 6.50730046317284448303947966062, 6.51210341109520376412783574737, 6.99088385583546517006611864781, 7.70736888774135246414550699110, 7.947221900702744398451124637937, 8.592230161283651090333120470135, 8.646583260804489178670566003207, 8.967798053409226243916777565384, 9.535540457390030055217666489754, 9.580522065062501171198349789066

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