Properties

Label 4-184e2-1.1-c1e2-0-1
Degree $4$
Conductor $33856$
Sign $1$
Analytic cond. $2.15868$
Root an. cond. $1.21212$
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 + 4·5-s − 9-s + 2·11-s + 5·13-s − 4·15-s + 2·17-s + 2·19-s − 2·23-s + 2·25-s + 3·29-s − 9·31-s − 2·33-s − 5·39-s + 41-s − 16·43-s − 4·45-s + 11·47-s − 14·49-s − 2·51-s + 4·53-s + 8·55-s − 2·57-s + 4·59-s + 8·61-s + 20·65-s − 2·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s − 1/3·9-s + 0.603·11-s + 1.38·13-s − 1.03·15-s + 0.485·17-s + 0.458·19-s − 0.417·23-s + 2/5·25-s + 0.557·29-s − 1.61·31-s − 0.348·33-s − 0.800·39-s + 0.156·41-s − 2.43·43-s − 0.596·45-s + 1.60·47-s − 2·49-s − 0.280·51-s + 0.549·53-s + 1.07·55-s − 0.264·57-s + 0.520·59-s + 1.02·61-s + 2.48·65-s − 0.244·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33856\)    =    \(2^{6} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(2.15868\)
Root analytic conductor: \(1.21212\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{184} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 33856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.482145850\)
\(L(\frac12)\) \(\approx\) \(1.482145850\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 11 T + 86 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 23 T + 270 T^{2} - 23 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−12.95027483806111710042162768931, −12.42166299949270058109746642113, −11.66929483163137957912417928418, −11.50360440708631699291947275935, −10.91275619474228902607669351037, −10.34573827236301225286728455732, −9.743313128799501782362351196014, −9.677018859273705375482052255716, −8.847535098094654825696978388046, −8.522477930042910374593748983406, −7.76853425449103474434565916150, −6.98921704429773607482381407808, −6.23872052932777940841616403118, −6.15229576525720049532558019035, −5.43391800843212204758872795626, −5.16498978398196052586095633282, −3.93675101810801057119345153741, −3.36426087235987012026418663320, −2.14326713709089251974870539732, −1.41817274069713329556021365021, 1.41817274069713329556021365021, 2.14326713709089251974870539732, 3.36426087235987012026418663320, 3.93675101810801057119345153741, 5.16498978398196052586095633282, 5.43391800843212204758872795626, 6.15229576525720049532558019035, 6.23872052932777940841616403118, 6.98921704429773607482381407808, 7.76853425449103474434565916150, 8.522477930042910374593748983406, 8.847535098094654825696978388046, 9.677018859273705375482052255716, 9.743313128799501782362351196014, 10.34573827236301225286728455732, 10.91275619474228902607669351037, 11.50360440708631699291947275935, 11.66929483163137957912417928418, 12.42166299949270058109746642113, 12.95027483806111710042162768931

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