Properties

Label 4-1148e2-1.1-c1e2-0-5
Degree $4$
Conductor $1317904$
Sign $1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
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 + 5-s + 4·7-s + 3·9-s − 3·11-s + 4·13-s + 15-s − 17-s + 3·19-s + 4·21-s + 5·23-s + 5·25-s + 8·27-s − 4·29-s − 5·31-s − 3·33-s + 4·35-s − 7·37-s + 4·39-s + 2·41-s + 8·43-s + 3·45-s + 3·47-s + 9·49-s − 51-s + 3·53-s − 3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s + 9-s − 0.904·11-s + 1.10·13-s + 0.258·15-s − 0.242·17-s + 0.688·19-s + 0.872·21-s + 1.04·23-s + 25-s + 1.53·27-s − 0.742·29-s − 0.898·31-s − 0.522·33-s + 0.676·35-s − 1.15·37-s + 0.640·39-s + 0.312·41-s + 1.21·43-s + 0.447·45-s + 0.437·47-s + 9/7·49-s − 0.140·51-s + 0.412·53-s − 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1148} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.500847835\)
\(L(\frac12)\) \(\approx\) \(4.500847835\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
41$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 5 T - 64 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−9.961008004776927866165771651760, −9.583494187759943191617791722417, −8.899752144273314320040223909686, −8.899485584939969874396445523537, −8.482352520140639186940626269576, −7.937867437248048029814234220065, −7.44021761200527431332599522526, −7.39858090564515469227861968815, −6.74128116068101919904724702578, −6.31836680348769194072082136327, −5.51213348401063149214960519876, −5.39258882848160249493338012395, −4.68075855893167676331833226928, −4.66216835570596692595637984487, −3.59820411267085500824298567338, −3.57165973902465274383172110991, −2.54398730826856149381154172190, −2.25941365632885050281541916607, −1.39101956254902179472605731523, −1.06437526557409089207291463670, 1.06437526557409089207291463670, 1.39101956254902179472605731523, 2.25941365632885050281541916607, 2.54398730826856149381154172190, 3.57165973902465274383172110991, 3.59820411267085500824298567338, 4.66216835570596692595637984487, 4.68075855893167676331833226928, 5.39258882848160249493338012395, 5.51213348401063149214960519876, 6.31836680348769194072082136327, 6.74128116068101919904724702578, 7.39858090564515469227861968815, 7.44021761200527431332599522526, 7.937867437248048029814234220065, 8.482352520140639186940626269576, 8.899485584939969874396445523537, 8.899752144273314320040223909686, 9.583494187759943191617791722417, 9.961008004776927866165771651760

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