Properties

Label 4-624e2-1.1-c0e2-0-3
Degree $4$
Conductor $389376$
Sign $1$
Analytic cond. $0.0969802$
Root an. cond. $0.558047$
Motivic weight $0$
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 + 3·7-s + 13-s − 3·21-s − 2·25-s + 27-s − 39-s − 43-s + 5·49-s − 61-s − 3·67-s + 2·75-s − 2·79-s − 81-s + 3·91-s − 3·97-s + 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s − 5·147-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 3-s + 3·7-s + 13-s − 3·21-s − 2·25-s + 27-s − 39-s − 43-s + 5·49-s − 61-s − 3·67-s + 2·75-s − 2·79-s − 81-s + 3·91-s − 3·97-s + 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s − 5·147-s + 149-s + 151-s + 157-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(389376\)    =    \(2^{8} \cdot 3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.0969802\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{624} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 389376,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8143423264\)
\(L(\frac12)\) \(\approx\) \(0.8143423264\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 - T + T^{2} \)
good5$C_2$ \( ( 1 + T^{2} )^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 - T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2^2$ \( 1 - T^{2} + T^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
71$C_2^2$ \( 1 - T^{2} + T^{4} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 + T + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2^2$ \( 1 - T^{2} + T^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T + T^{2} ) \)
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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

−11.16326648644933549402327645131, −10.83262064549757261779025819985, −10.29586929035861436136625478662, −9.998511309693193538274555644499, −9.176867286509257525104333490791, −8.668807895706193204494562136858, −8.390855695567502536544133074677, −8.065784939512042983425540800756, −7.39264452533451562382393925326, −7.37301608915810004823649888907, −6.32871874525778602190014383538, −5.97583254475485776225850655691, −5.54484731255588674475720137007, −5.15284744389746398276964585931, −4.46788908484202529413591837651, −4.41813693880371658290269596008, −3.57909551619564285295912477576, −2.61655853986043007549936872830, −1.59513767482707627453932159978, −1.50515612135362487738805585777, 1.50515612135362487738805585777, 1.59513767482707627453932159978, 2.61655853986043007549936872830, 3.57909551619564285295912477576, 4.41813693880371658290269596008, 4.46788908484202529413591837651, 5.15284744389746398276964585931, 5.54484731255588674475720137007, 5.97583254475485776225850655691, 6.32871874525778602190014383538, 7.37301608915810004823649888907, 7.39264452533451562382393925326, 8.065784939512042983425540800756, 8.390855695567502536544133074677, 8.668807895706193204494562136858, 9.176867286509257525104333490791, 9.998511309693193538274555644499, 10.29586929035861436136625478662, 10.83262064549757261779025819985, 11.16326648644933549402327645131

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