Properties

Label 4-700e2-1.1-c1e2-0-1
Degree $4$
Conductor $490000$
Sign $1$
Analytic cond. $31.2428$
Root an. cond. $2.36421$
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-s − 5·7-s + 3·9-s − 6·11-s − 4·13-s − 6·17-s − 8·19-s − 5·21-s + 3·23-s + 8·27-s + 6·29-s − 2·31-s − 6·33-s + 8·37-s − 4·39-s − 6·41-s − 10·43-s + 18·49-s − 6·51-s + 12·53-s − 8·57-s + 61-s − 15·63-s − 7·67-s + 3·69-s − 10·73-s + 30·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.88·7-s + 9-s − 1.80·11-s − 1.10·13-s − 1.45·17-s − 1.83·19-s − 1.09·21-s + 0.625·23-s + 1.53·27-s + 1.11·29-s − 0.359·31-s − 1.04·33-s + 1.31·37-s − 0.640·39-s − 0.937·41-s − 1.52·43-s + 18/7·49-s − 0.840·51-s + 1.64·53-s − 1.05·57-s + 0.128·61-s − 1.88·63-s − 0.855·67-s + 0.361·69-s − 1.17·73-s + 3.41·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(490000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.2428\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{700} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 490000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7343845202\)
\(L(\frac12)\) \(\approx\) \(0.7343845202\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−10.41362188078514003608359287354, −10.25953784270951621519398279311, −10.00324023499322021332252933782, −9.337261374038248347063379468643, −8.918974111919296854243435642879, −8.658489491408132230019815303291, −8.087215895227938877152555156762, −7.62258246659032007455583264361, −7.02946516169490005460335523496, −6.65433293208522760385499514295, −6.57626711563376626277339237689, −5.79519194074867343360261129083, −5.16274023227331170122833576164, −4.51969856602132820936361931975, −4.37290623962142660537668212436, −3.51753398569155822286679401121, −2.74985629227441079877287027896, −2.67440762846505507262304340718, −2.00191864766087194181105551217, −0.40420496194072384370824523304, 0.40420496194072384370824523304, 2.00191864766087194181105551217, 2.67440762846505507262304340718, 2.74985629227441079877287027896, 3.51753398569155822286679401121, 4.37290623962142660537668212436, 4.51969856602132820936361931975, 5.16274023227331170122833576164, 5.79519194074867343360261129083, 6.57626711563376626277339237689, 6.65433293208522760385499514295, 7.02946516169490005460335523496, 7.62258246659032007455583264361, 8.087215895227938877152555156762, 8.658489491408132230019815303291, 8.918974111919296854243435642879, 9.337261374038248347063379468643, 10.00324023499322021332252933782, 10.25953784270951621519398279311, 10.41362188078514003608359287354

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