Properties

Label 4-980000-1.1-c1e2-0-8
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 5·9-s + 4·13-s + 16-s + 6·17-s − 5·18-s + 4·26-s − 12·29-s + 32-s + 6·34-s − 5·36-s + 16·37-s − 18·41-s + 49-s + 4·52-s − 24·53-s − 12·58-s − 20·61-s + 64-s + 6·68-s − 5·72-s + 10·73-s + 16·74-s + 16·81-s − 18·82-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 5/3·9-s + 1.10·13-s + 1/4·16-s + 1.45·17-s − 1.17·18-s + 0.784·26-s − 2.22·29-s + 0.176·32-s + 1.02·34-s − 5/6·36-s + 2.63·37-s − 2.81·41-s + 1/7·49-s + 0.554·52-s − 3.29·53-s − 1.57·58-s − 2.56·61-s + 1/8·64-s + 0.727·68-s − 0.589·72-s + 1.17·73-s + 1.85·74-s + 16/9·81-s − 1.98·82-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: \(1\)
Selberg data: \((4,\ 980000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
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

−7.77658810173983217795932841222, −7.76136885452014769391786338919, −6.84082843689233258487775944714, −6.50150023662226984648693008298, −5.92654096551589702125781476334, −5.63302791327341414299631153495, −5.48776820834083428423049797351, −4.68405477142379205905809833243, −4.26850617197803948832200761616, −3.37835476194466045277789097411, −3.32408442847114247075118763103, −2.85425374545583595819763274865, −1.89984331795161626562592295843, −1.33133249048010073515633278194, 0, 1.33133249048010073515633278194, 1.89984331795161626562592295843, 2.85425374545583595819763274865, 3.32408442847114247075118763103, 3.37835476194466045277789097411, 4.26850617197803948832200761616, 4.68405477142379205905809833243, 5.48776820834083428423049797351, 5.63302791327341414299631153495, 5.92654096551589702125781476334, 6.50150023662226984648693008298, 6.84082843689233258487775944714, 7.76136885452014769391786338919, 7.77658810173983217795932841222

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