Properties

Label 4-624e2-1.1-c1e2-0-36
Degree $4$
Conductor $389376$
Sign $1$
Analytic cond. $24.8269$
Root an. cond. $2.23218$
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  + 2·3-s + 3·9-s − 2·13-s − 12·17-s + 10·25-s + 4·27-s + 12·29-s − 4·39-s + 8·43-s + 2·49-s − 24·51-s + 12·53-s − 4·61-s + 20·75-s + 16·79-s + 5·81-s + 24·87-s + 12·101-s − 16·103-s − 24·107-s − 12·113-s − 6·117-s + 10·121-s + 127-s + 16·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 0.554·13-s − 2.91·17-s + 2·25-s + 0.769·27-s + 2.22·29-s − 0.640·39-s + 1.21·43-s + 2/7·49-s − 3.36·51-s + 1.64·53-s − 0.512·61-s + 2.30·75-s + 1.80·79-s + 5/9·81-s + 2.57·87-s + 1.19·101-s − 1.57·103-s − 2.32·107-s − 1.12·113-s − 0.554·117-s + 0.909·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(389376\)    =    \(2^{8} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(24.8269\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 389376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.682293213\)
\(L(\frac12)\) \(\approx\) \(2.682293213\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.75689756272077707582952716091, −10.66319121561289677400757188784, −9.686147721688605379665328424707, −9.562311045310686475041676405335, −8.935791168218723743976795167051, −8.576641639165694473556274004970, −8.510765237236881745588180510578, −7.86718136373754710577523334846, −7.08042949451288875981485634498, −6.99088465860776900368203948303, −6.56082946455080767729923637818, −6.00493265352164549466872659514, −5.07844470027483760381842639919, −4.64550640467252958945267459081, −4.36860633865166102426951112563, −3.73598920899971406734335314969, −2.75771868111859403138891802347, −2.66075473752642768838636405485, −2.00657706072832352684294701813, −0.860671078871469360885599871875, 0.860671078871469360885599871875, 2.00657706072832352684294701813, 2.66075473752642768838636405485, 2.75771868111859403138891802347, 3.73598920899971406734335314969, 4.36860633865166102426951112563, 4.64550640467252958945267459081, 5.07844470027483760381842639919, 6.00493265352164549466872659514, 6.56082946455080767729923637818, 6.99088465860776900368203948303, 7.08042949451288875981485634498, 7.86718136373754710577523334846, 8.510765237236881745588180510578, 8.576641639165694473556274004970, 8.935791168218723743976795167051, 9.562311045310686475041676405335, 9.686147721688605379665328424707, 10.66319121561289677400757188784, 10.75689756272077707582952716091

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