Properties

Label 4-630e2-1.1-c1e2-0-29
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
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-s − 5-s − 4·7-s − 8-s − 10-s + 3·11-s + 10·13-s − 4·14-s − 16-s + 6·17-s + 19-s + 3·22-s + 3·23-s + 10·26-s + 12·29-s + 4·31-s + 6·34-s + 4·35-s − 11·37-s + 38-s + 40-s − 6·41-s − 20·43-s + 3·46-s + 3·47-s + 9·49-s + 3·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.904·11-s + 2.77·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.639·22-s + 0.625·23-s + 1.96·26-s + 2.22·29-s + 0.718·31-s + 1.02·34-s + 0.676·35-s − 1.80·37-s + 0.162·38-s + 0.158·40-s − 0.937·41-s − 3.04·43-s + 0.442·46-s + 0.437·47-s + 9/7·49-s + 0.412·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.479911251\)
\(L(\frac12)\) \(\approx\) \(2.479911251\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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

−10.56419179604024261429274951854, −10.44818051785564739078228122799, −10.12612932223358285627844473903, −9.502503365720733073728977407072, −8.905848776088895795899535002048, −8.744832402103173525786839833692, −8.279904560490615960648860263796, −7.85243620877010769439574829020, −6.89761810652307725636640280718, −6.70694526413765678264311412875, −6.23591505780769908319746527635, −6.10542688645873288368260961912, −5.14995645061735800448202820056, −4.99887129157353311866874898179, −3.87479515698245061653802027953, −3.76805112712822206061347838175, −3.24024604742150131386801335067, −3.02350979959323521801885611695, −1.53824127778636840542222600082, −0.871158682691370579625836974050, 0.871158682691370579625836974050, 1.53824127778636840542222600082, 3.02350979959323521801885611695, 3.24024604742150131386801335067, 3.76805112712822206061347838175, 3.87479515698245061653802027953, 4.99887129157353311866874898179, 5.14995645061735800448202820056, 6.10542688645873288368260961912, 6.23591505780769908319746527635, 6.70694526413765678264311412875, 6.89761810652307725636640280718, 7.85243620877010769439574829020, 8.279904560490615960648860263796, 8.744832402103173525786839833692, 8.905848776088895795899535002048, 9.502503365720733073728977407072, 10.12612932223358285627844473903, 10.44818051785564739078228122799, 10.56419179604024261429274951854

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