Properties

Label 4-2736e2-1.1-c1e2-0-32
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $477.294$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 6·7-s − 5·13-s − 4·17-s + 8·19-s − 6·23-s + 5·25-s − 4·29-s + 14·31-s + 12·35-s − 2·37-s + 11·43-s − 6·47-s + 13·49-s − 2·53-s − 8·59-s − 7·61-s − 10·65-s + 3·67-s − 6·71-s − 9·73-s + 5·79-s + 32·83-s − 8·85-s + 12·89-s − 30·91-s + 16·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 2.26·7-s − 1.38·13-s − 0.970·17-s + 1.83·19-s − 1.25·23-s + 25-s − 0.742·29-s + 2.51·31-s + 2.02·35-s − 0.328·37-s + 1.67·43-s − 0.875·47-s + 13/7·49-s − 0.274·53-s − 1.04·59-s − 0.896·61-s − 1.24·65-s + 0.366·67-s − 0.712·71-s − 1.05·73-s + 0.562·79-s + 3.51·83-s − 0.867·85-s + 1.27·89-s − 3.14·91-s + 1.64·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(477.294\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.199188062\)
\(L(\frac12)\) \(\approx\) \(4.199188062\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−9.281968304190846881382656072117, −8.511535382787352037929656385904, −8.056498021957868892901208731156, −8.034341938820332371636715427319, −7.54427566591219358721419870725, −7.29647068910001796852401970758, −6.74792265808836219270128844740, −6.20876152325514070858790975166, −6.01513035474501692563058594377, −5.36491778444830718988827758542, −4.97222143434921049825181007593, −4.88968745443964316322907829406, −4.47351210385094920700078117889, −4.03989490455344729230709226881, −3.21976151845715936982518385080, −2.72277130391418595595419827095, −2.29204719660334904828069524856, −1.79780884082885014557631323343, −1.42155051377593773348860172898, −0.67347671895627927502237874484, 0.67347671895627927502237874484, 1.42155051377593773348860172898, 1.79780884082885014557631323343, 2.29204719660334904828069524856, 2.72277130391418595595419827095, 3.21976151845715936982518385080, 4.03989490455344729230709226881, 4.47351210385094920700078117889, 4.88968745443964316322907829406, 4.97222143434921049825181007593, 5.36491778444830718988827758542, 6.01513035474501692563058594377, 6.20876152325514070858790975166, 6.74792265808836219270128844740, 7.29647068910001796852401970758, 7.54427566591219358721419870725, 8.034341938820332371636715427319, 8.056498021957868892901208731156, 8.511535382787352037929656385904, 9.281968304190846881382656072117

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