Properties

Label 4-1318707-1.1-c1e2-0-2
Degree $4$
Conductor $1318707$
Sign $1$
Analytic cond. $84.0819$
Root an. cond. $3.02813$
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  − 3-s − 3·4-s + 4·7-s + 9-s + 3·12-s − 2·13-s + 5·16-s + 8·19-s − 4·21-s + 6·25-s − 27-s − 12·28-s + 20·31-s − 3·36-s − 8·37-s + 2·39-s + 24·43-s − 5·48-s − 2·49-s + 6·52-s − 8·57-s − 20·61-s + 4·63-s − 3·64-s + 24·67-s + 8·73-s − 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 1.51·7-s + 1/3·9-s + 0.866·12-s − 0.554·13-s + 5/4·16-s + 1.83·19-s − 0.872·21-s + 6/5·25-s − 0.192·27-s − 2.26·28-s + 3.59·31-s − 1/2·36-s − 1.31·37-s + 0.320·39-s + 3.65·43-s − 0.721·48-s − 2/7·49-s + 0.832·52-s − 1.05·57-s − 2.56·61-s + 0.503·63-s − 3/8·64-s + 2.93·67-s + 0.936·73-s − 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1318707 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1318707 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1318707\)    =    \(3^{3} \cdot 13^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(84.0819\)
Root analytic conductor: \(3.02813\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1318707,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.780059987\)
\(L(\frac12)\) \(\approx\) \(1.780059987\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 + T \)
13$C_1$ \( ( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.947679690284131595849776733076, −7.81342569999435272878515329062, −6.98832239970943095443656552142, −6.91261710272029920121825676613, −5.85102904238401179169986596678, −5.84386084483901740500080157566, −5.05964754688132425608205366177, −4.73182091516976951232105158648, −4.68717336447753969967438730348, −4.18452058198540408772009248966, −3.35760137954518791100219081623, −2.87743116741757128615804592887, −2.06769892169864904494042186723, −0.955175571715672715988306088693, −0.905158356090526027105523059946, 0.905158356090526027105523059946, 0.955175571715672715988306088693, 2.06769892169864904494042186723, 2.87743116741757128615804592887, 3.35760137954518791100219081623, 4.18452058198540408772009248966, 4.68717336447753969967438730348, 4.73182091516976951232105158648, 5.05964754688132425608205366177, 5.84386084483901740500080157566, 5.85102904238401179169986596678, 6.91261710272029920121825676613, 6.98832239970943095443656552142, 7.81342569999435272878515329062, 7.947679690284131595849776733076

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