Properties

Label 4-980000-1.1-c1e2-0-7
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 2·9-s + 8·13-s + 16-s − 12·17-s − 2·18-s + 8·26-s − 12·29-s + 32-s − 12·34-s − 2·36-s − 4·37-s + 12·41-s + 49-s + 8·52-s − 12·53-s − 12·58-s + 16·61-s + 64-s − 12·68-s − 2·72-s − 4·73-s − 4·74-s − 5·81-s + 12·82-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2/3·9-s + 2.21·13-s + 1/4·16-s − 2.91·17-s − 0.471·18-s + 1.56·26-s − 2.22·29-s + 0.176·32-s − 2.05·34-s − 1/3·36-s − 0.657·37-s + 1.87·41-s + 1/7·49-s + 1.10·52-s − 1.64·53-s − 1.57·58-s + 2.04·61-s + 1/8·64-s − 1.45·68-s − 0.235·72-s − 0.468·73-s − 0.464·74-s − 5/9·81-s + 1.32·82-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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.71895664384636679391588711627, −7.60707532453471338039224955289, −6.66965130292041403123447657278, −6.60348506563741211247253098956, −6.19185902002877876721827038865, −5.59857578974307577358363282055, −5.38990980084792703269563412867, −4.58557293909771964767836479942, −4.11500666640753372808858013263, −3.83664653997120963252560991499, −3.28327332646103144207952017478, −2.50419401332512614244886579026, −2.07204529193989626181377807359, −1.31920416816215806650160602271, 0, 1.31920416816215806650160602271, 2.07204529193989626181377807359, 2.50419401332512614244886579026, 3.28327332646103144207952017478, 3.83664653997120963252560991499, 4.11500666640753372808858013263, 4.58557293909771964767836479942, 5.38990980084792703269563412867, 5.59857578974307577358363282055, 6.19185902002877876721827038865, 6.60348506563741211247253098956, 6.66965130292041403123447657278, 7.60707532453471338039224955289, 7.71895664384636679391588711627

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