Properties

Label 4-980000-1.1-c1e2-0-2
Degree $4$
Conductor $980000$
Sign $1$
Analytic cond. $62.4856$
Root an. cond. $2.81154$
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 + 4-s − 8-s − 6·9-s + 12·13-s + 16-s − 4·17-s + 6·18-s − 12·26-s + 12·29-s − 32-s + 4·34-s − 6·36-s + 20·37-s + 4·41-s + 49-s + 12·52-s + 4·53-s − 12·58-s − 28·61-s + 64-s − 4·68-s + 6·72-s − 4·73-s − 20·74-s + 27·81-s − 4·82-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2·9-s + 3.32·13-s + 1/4·16-s − 0.970·17-s + 1.41·18-s − 2.35·26-s + 2.22·29-s − 0.176·32-s + 0.685·34-s − 36-s + 3.28·37-s + 0.624·41-s + 1/7·49-s + 1.66·52-s + 0.549·53-s − 1.57·58-s − 3.58·61-s + 1/8·64-s − 0.485·68-s + 0.707·72-s − 0.468·73-s − 2.32·74-s + 3·81-s − 0.441·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(980000\)    =    \(2^{5} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(62.4856\)
Root analytic conductor: \(2.81154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{980000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 980000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.479979573\)
\(L(\frac12)\) \(\approx\) \(1.479979573\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.107288137075358285056657641589, −8.089729535357517933440155073370, −7.41297246540534830906507852807, −6.56718920518778472538358027386, −6.24190347389181960956817365315, −6.01430128602422585899125610915, −5.91578381006435814657965447568, −4.99516684089165156540912778411, −4.36823223624486251008730105784, −3.94248355024955497277318994877, −3.07578209230613039021513049561, −2.98711940319117381089884417433, −2.26639998875518159229729259543, −1.27008084899723889248060652436, −0.71428970411115330695948810032, 0.71428970411115330695948810032, 1.27008084899723889248060652436, 2.26639998875518159229729259543, 2.98711940319117381089884417433, 3.07578209230613039021513049561, 3.94248355024955497277318994877, 4.36823223624486251008730105784, 4.99516684089165156540912778411, 5.91578381006435814657965447568, 6.01430128602422585899125610915, 6.24190347389181960956817365315, 6.56718920518778472538358027386, 7.41297246540534830906507852807, 8.089729535357517933440155073370, 8.107288137075358285056657641589

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