Properties

Label 4-1000188-1.1-c1e2-0-20
Degree $4$
Conductor $1000188$
Sign $-1$
Analytic cond. $63.7728$
Root an. cond. $2.82591$
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  + 4-s − 2·5-s − 7-s + 16-s − 4·17-s − 2·20-s − 7·25-s − 28-s + 2·35-s + 10·37-s + 18·41-s − 20·43-s − 12·47-s + 49-s + 28·59-s + 64-s − 16·67-s − 4·68-s + 12·79-s − 2·80-s + 8·83-s + 8·85-s + 18·89-s − 7·100-s + 28·101-s + 14·109-s − 112-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.894·5-s − 0.377·7-s + 1/4·16-s − 0.970·17-s − 0.447·20-s − 7/5·25-s − 0.188·28-s + 0.338·35-s + 1.64·37-s + 2.81·41-s − 3.04·43-s − 1.75·47-s + 1/7·49-s + 3.64·59-s + 1/8·64-s − 1.95·67-s − 0.485·68-s + 1.35·79-s − 0.223·80-s + 0.878·83-s + 0.867·85-s + 1.90·89-s − 0.699·100-s + 2.78·101-s + 1.34·109-s − 0.0944·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000188 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000188 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1000188\)    =    \(2^{2} \cdot 3^{6} \cdot 7^{3}\)
Sign: $-1$
Analytic conductor: \(63.7728\)
Root analytic conductor: \(2.82591\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1000188} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1000188,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
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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

−8.015044568391676191880321656496, −7.43396451799616319962951174764, −7.15629851020103523042213387688, −6.47784583599995832693494924500, −6.23227538486956081778076113568, −5.87450213364309522501266401702, −5.08207234913297047777508467306, −4.69069334189536458166730202416, −4.15202289907072344630083460584, −3.57313298834818934312909000630, −3.35511404251805033708538922757, −2.26551691900571842895890768283, −2.24049119020186278469176393363, −1.01936668222152272292011651270, 0, 1.01936668222152272292011651270, 2.24049119020186278469176393363, 2.26551691900571842895890768283, 3.35511404251805033708538922757, 3.57313298834818934312909000630, 4.15202289907072344630083460584, 4.69069334189536458166730202416, 5.08207234913297047777508467306, 5.87450213364309522501266401702, 6.23227538486956081778076113568, 6.47784583599995832693494924500, 7.15629851020103523042213387688, 7.43396451799616319962951174764, 8.015044568391676191880321656496

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