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Properties

Label 4-855e2-1.1-c1e2-0-14
Degree $4$
Conductor $731025$
Sign $1$
Analytic cond. $46.6107$
Root an. cond. $2.61289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 4·5-s + 4·11-s + 5·16-s − 2·19-s + 12·20-s + 11·25-s + 12·29-s − 8·31-s + 4·41-s + 12·44-s + 10·49-s + 16·55-s − 20·61-s + 3·64-s − 16·71-s − 6·76-s + 16·79-s + 20·80-s − 20·89-s − 8·95-s + 33·100-s + 12·109-s + 36·116-s − 10·121-s − 24·124-s + 24·125-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.78·5-s + 1.20·11-s + 5/4·16-s − 0.458·19-s + 2.68·20-s + 11/5·25-s + 2.22·29-s − 1.43·31-s + 0.624·41-s + 1.80·44-s + 10/7·49-s + 2.15·55-s − 2.56·61-s + 3/8·64-s − 1.89·71-s − 0.688·76-s + 1.80·79-s + 2.23·80-s − 2.11·89-s − 0.820·95-s + 3.29·100-s + 1.14·109-s + 3.34·116-s − 0.909·121-s − 2.15·124-s + 2.14·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(731025\)    =    \(3^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(46.6107\)
Root analytic conductor: \(2.61289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 731025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.009034743\)
\(L(\frac12)\) \(\approx\) \(5.009034743\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad3 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−10.41737247471285469807786476730, −10.12646617100499277098909120431, −9.483567440917630241018606292112, −9.250840009713004995148678350012, −8.754137495052402914892306546719, −8.479230701203551187422644813196, −7.56024957916006174241903721531, −7.38306704759315663432768293714, −6.72892563003714813428130132180, −6.51362987178189988334855362282, −5.99831679603070817721854291951, −5.96564902743805933963082980578, −5.20257039028099728325110258959, −4.67836198328781943963638737601, −4.03065092351358301673726484637, −3.30432615157517333132330895532, −2.56970910005043211489291031175, −2.44436305421132028083560688713, −1.50865096095262405934657522825, −1.31120513869035294460168071989, 1.31120513869035294460168071989, 1.50865096095262405934657522825, 2.44436305421132028083560688713, 2.56970910005043211489291031175, 3.30432615157517333132330895532, 4.03065092351358301673726484637, 4.67836198328781943963638737601, 5.20257039028099728325110258959, 5.96564902743805933963082980578, 5.99831679603070817721854291951, 6.51362987178189988334855362282, 6.72892563003714813428130132180, 7.38306704759315663432768293714, 7.56024957916006174241903721531, 8.479230701203551187422644813196, 8.754137495052402914892306546719, 9.250840009713004995148678350012, 9.483567440917630241018606292112, 10.12646617100499277098909120431, 10.41737247471285469807786476730

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