Properties

Label 4-1216e2-1.1-c1e2-0-24
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 2·9-s + 8·13-s − 6·17-s + 17·25-s − 12·29-s − 4·37-s − 12·41-s + 12·45-s − 13·49-s − 24·53-s + 2·61-s − 48·65-s − 14·73-s − 5·81-s + 36·85-s + 24·89-s + 16·97-s − 12·101-s + 32·109-s + 12·113-s − 16·117-s − 13·121-s − 18·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 2.68·5-s − 2/3·9-s + 2.21·13-s − 1.45·17-s + 17/5·25-s − 2.22·29-s − 0.657·37-s − 1.87·41-s + 1.78·45-s − 1.85·49-s − 3.29·53-s + 0.256·61-s − 5.95·65-s − 1.63·73-s − 5/9·81-s + 3.90·85-s + 2.54·89-s + 1.62·97-s − 1.19·101-s + 3.06·109-s + 1.12·113-s − 1.47·117-s − 1.18·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1478656} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−7.65894969899749366452649043261, −7.15811652768508964807251071197, −6.68601933176938591059893570713, −6.07700564140158959836320975863, −6.05813289271022268292942638231, −4.96903618429312636799150610379, −4.80419327140120764096716079767, −4.19306407725294924458030580762, −3.67310116806022314895429635564, −3.33866387034377157662854676826, −3.31935920587958999413340629739, −2.04259190544357015606449170426, −1.39430917521719853589747801849, 0, 0, 1.39430917521719853589747801849, 2.04259190544357015606449170426, 3.31935920587958999413340629739, 3.33866387034377157662854676826, 3.67310116806022314895429635564, 4.19306407725294924458030580762, 4.80419327140120764096716079767, 4.96903618429312636799150610379, 6.05813289271022268292942638231, 6.07700564140158959836320975863, 6.68601933176938591059893570713, 7.15811652768508964807251071197, 7.65894969899749366452649043261

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