Properties

Label 4-1305e2-1.1-c1e2-0-12
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  − 2-s − 2·4-s − 2·5-s − 4·7-s + 3·8-s + 2·10-s + 4·11-s + 2·13-s + 4·14-s + 16-s + 6·17-s − 4·19-s + 4·20-s − 4·22-s + 3·25-s − 2·26-s + 8·28-s − 2·29-s − 16·31-s − 2·32-s − 6·34-s + 8·35-s + 4·38-s − 6·40-s − 12·41-s − 12·43-s − 8·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s − 0.894·5-s − 1.51·7-s + 1.06·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.894·20-s − 0.852·22-s + 3/5·25-s − 0.392·26-s + 1.51·28-s − 0.371·29-s − 2.87·31-s − 0.353·32-s − 1.02·34-s + 1.35·35-s + 0.648·38-s − 0.948·40-s − 1.87·41-s − 1.82·43-s − 1.20·44-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$C_1$ \( ( 1 + T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 12 T + 125 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 16 T + 153 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−9.239748993847396031045299579043, −9.001685557717831671157541948911, −8.671828787075929961567890920991, −8.653453618454288766397376057688, −7.68589258403912868889616635713, −7.55075159942762812595657162738, −7.19652304162427331253874108891, −6.56296338581829909620787185137, −6.08738029440758045555188525027, −5.93066831115428455198880535892, −5.10215716093478256675803103231, −4.74585782822978976758725290042, −4.07767200831033873658644187654, −3.71842405262805641037485466416, −3.39296885699055149250019936757, −3.05785282940764957868027393824, −1.74876433208676821022954593604, −1.28568299950952581982165489920, 0, 0, 1.28568299950952581982165489920, 1.74876433208676821022954593604, 3.05785282940764957868027393824, 3.39296885699055149250019936757, 3.71842405262805641037485466416, 4.07767200831033873658644187654, 4.74585782822978976758725290042, 5.10215716093478256675803103231, 5.93066831115428455198880535892, 6.08738029440758045555188525027, 6.56296338581829909620787185137, 7.19652304162427331253874108891, 7.55075159942762812595657162738, 7.68589258403912868889616635713, 8.653453618454288766397376057688, 8.671828787075929961567890920991, 9.001685557717831671157541948911, 9.239748993847396031045299579043

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