Properties

Label 4-1400e2-1.1-c1e2-0-9
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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  + 2·3-s − 7-s + 3·9-s − 4·11-s + 4·13-s − 3·17-s − 2·21-s + 3·23-s + 10·27-s − 12·29-s − 9·31-s − 8·33-s + 8·39-s + 10·41-s + 12·43-s + 9·47-s − 6·49-s − 6·51-s − 6·53-s − 8·59-s − 8·61-s − 3·63-s + 14·67-s + 6·69-s + 22·71-s − 2·73-s + 4·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 9-s − 1.20·11-s + 1.10·13-s − 0.727·17-s − 0.436·21-s + 0.625·23-s + 1.92·27-s − 2.22·29-s − 1.61·31-s − 1.39·33-s + 1.28·39-s + 1.56·41-s + 1.82·43-s + 1.31·47-s − 6/7·49-s − 0.840·51-s − 0.824·53-s − 1.04·59-s − 1.02·61-s − 0.377·63-s + 1.71·67-s + 0.722·69-s + 2.61·71-s − 0.234·73-s + 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.907529322\)
\(L(\frac12)\) \(\approx\) \(2.907529322\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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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

−9.442230302341272876940673159998, −9.316402721314430828805157942795, −8.911438316034635997633731928727, −8.828492515546851920530589745748, −7.963149347942184417149369613609, −7.88581045624189820291944312690, −7.42881538663985532489250437751, −7.13921398510323406508806801667, −6.48802365056284018082708748710, −6.10610431854181666728485279323, −5.65265750840310458565433198325, −5.13849713972030019075298364589, −4.65881410784362455438298351545, −4.06024491269712619656586931581, −3.59879437782135210841045002144, −3.33129490705527477394992038943, −2.53047616064698233314009098492, −2.31856820978026890140390153273, −1.58717608591610793170014415323, −0.64256761870295660791180051862, 0.64256761870295660791180051862, 1.58717608591610793170014415323, 2.31856820978026890140390153273, 2.53047616064698233314009098492, 3.33129490705527477394992038943, 3.59879437782135210841045002144, 4.06024491269712619656586931581, 4.65881410784362455438298351545, 5.13849713972030019075298364589, 5.65265750840310458565433198325, 6.10610431854181666728485279323, 6.48802365056284018082708748710, 7.13921398510323406508806801667, 7.42881538663985532489250437751, 7.88581045624189820291944312690, 7.963149347942184417149369613609, 8.828492515546851920530589745748, 8.911438316034635997633731928727, 9.316402721314430828805157942795, 9.442230302341272876940673159998

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