Properties

Label 4-76e3-1.1-c1e2-0-7
Degree $4$
Conductor $438976$
Sign $-1$
Analytic cond. $27.9894$
Root an. cond. $2.30011$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2·9-s + 6·11-s + 8·13-s + 4·16-s − 6·17-s + 19-s − 25-s − 12·29-s + 8·31-s + 4·36-s − 4·37-s − 2·43-s − 12·44-s − 13·49-s − 16·52-s − 24·53-s − 8·64-s + 12·68-s − 12·71-s − 14·73-s − 2·76-s − 16·79-s − 5·81-s + 24·83-s − 12·99-s + 2·100-s + ⋯
L(s)  = 1  − 4-s − 2/3·9-s + 1.80·11-s + 2.21·13-s + 16-s − 1.45·17-s + 0.229·19-s − 1/5·25-s − 2.22·29-s + 1.43·31-s + 2/3·36-s − 0.657·37-s − 0.304·43-s − 1.80·44-s − 1.85·49-s − 2.21·52-s − 3.29·53-s − 64-s + 1.45·68-s − 1.42·71-s − 1.63·73-s − 0.229·76-s − 1.80·79-s − 5/9·81-s + 2.63·83-s − 1.20·99-s + 1/5·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 438976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 438976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(438976\)    =    \(2^{6} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(27.9894\)
Root analytic conductor: \(2.30011\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{438976} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 438976,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
19$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−8.462914393396165009068265832341, −8.051705452287886353637122667380, −7.65894969899749366452649043261, −6.68601933176938591059893570713, −6.51423862829933785566955564641, −6.07700564140158959836320975863, −5.66028747999011565064411683968, −4.96903618429312636799150610379, −4.20769275006497966985529904085, −4.19306407725294924458030580762, −3.33866387034377157662854676826, −3.17065949281414024554826462085, −1.68756739850759821134015575431, −1.39430917521719853589747801849, 0, 1.39430917521719853589747801849, 1.68756739850759821134015575431, 3.17065949281414024554826462085, 3.33866387034377157662854676826, 4.19306407725294924458030580762, 4.20769275006497966985529904085, 4.96903618429312636799150610379, 5.66028747999011565064411683968, 6.07700564140158959836320975863, 6.51423862829933785566955564641, 6.68601933176938591059893570713, 7.65894969899749366452649043261, 8.051705452287886353637122667380, 8.462914393396165009068265832341

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