Properties

Label 4-924e2-1.1-c1e2-0-30
Degree $4$
Conductor $853776$
Sign $-1$
Analytic cond. $54.4374$
Root an. cond. $2.71628$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 6·5-s + 2·7-s − 2·9-s − 6·15-s + 12·17-s + 2·21-s + 17·25-s − 5·27-s − 12·35-s − 2·37-s − 20·43-s + 12·45-s − 3·49-s + 12·51-s + 6·59-s − 4·63-s − 2·67-s + 17·75-s + 4·79-s + 81-s + 12·83-s − 72·85-s − 18·89-s + 36·101-s − 12·105-s + 4·109-s + ⋯
L(s)  = 1  + 0.577·3-s − 2.68·5-s + 0.755·7-s − 2/3·9-s − 1.54·15-s + 2.91·17-s + 0.436·21-s + 17/5·25-s − 0.962·27-s − 2.02·35-s − 0.328·37-s − 3.04·43-s + 1.78·45-s − 3/7·49-s + 1.68·51-s + 0.781·59-s − 0.503·63-s − 0.244·67-s + 1.96·75-s + 0.450·79-s + 1/9·81-s + 1.31·83-s − 7.80·85-s − 1.90·89-s + 3.58·101-s − 1.17·105-s + 0.383·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 853776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(853776\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(54.4374\)
Root analytic conductor: \(2.71628\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 853776,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 - T + p T^{2} \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.030135611649600137523792877956, −7.60852684015633966980596452192, −7.57107715370897119251299700280, −6.85141867351762882928486006631, −6.29465577828163423751170725006, −5.55064102283955097263086227457, −5.11227132195889762323685903220, −4.80957445265205313059818308287, −3.98043613480578289233430491973, −3.69194524730832991586322636375, −3.27381985666804460223319703873, −3.00610279743917756108045023038, −1.87692294389813555618766341906, −1.00356671220607536314623203552, 0, 1.00356671220607536314623203552, 1.87692294389813555618766341906, 3.00610279743917756108045023038, 3.27381985666804460223319703873, 3.69194524730832991586322636375, 3.98043613480578289233430491973, 4.80957445265205313059818308287, 5.11227132195889762323685903220, 5.55064102283955097263086227457, 6.29465577828163423751170725006, 6.85141867351762882928486006631, 7.57107715370897119251299700280, 7.60852684015633966980596452192, 8.030135611649600137523792877956

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