Properties

Label 4-1216e2-1.1-c1e2-0-3
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 $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s + 3·9-s + 6·11-s + 2·13-s + 4·15-s − 2·17-s + 19-s + 6·23-s + 5·25-s − 8·27-s − 4·29-s + 20·31-s − 6·33-s − 4·37-s − 2·39-s − 9·41-s + 4·43-s − 12·45-s − 12·47-s − 14·49-s + 2·51-s − 2·53-s − 24·55-s − 57-s + 59-s − 8·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 9-s + 1.80·11-s + 0.554·13-s + 1.03·15-s − 0.485·17-s + 0.229·19-s + 1.25·23-s + 25-s − 1.53·27-s − 0.742·29-s + 3.59·31-s − 1.04·33-s − 0.657·37-s − 0.320·39-s − 1.40·41-s + 0.609·43-s − 1.78·45-s − 1.75·47-s − 2·49-s + 0.280·51-s − 0.274·53-s − 3.23·55-s − 0.132·57-s + 0.130·59-s − 1.02·61-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: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.233402106\)
\(L(\frac12)\) \(\approx\) \(1.233402106\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.19047343942992710068432091141, −9.588603199068747606250465893550, −8.948549508496821513849176904799, −8.788506542317774987935002109008, −8.287773417794300723168477318689, −7.70415298038578573845057256038, −7.64794508068527693644007941078, −6.91176462577580700595313155801, −6.73075461733201898288233715476, −6.18918864564920868474958882471, −6.06649326926350063125449392390, −4.84266368455744552217499572239, −4.83491829718995064412919922657, −4.36370824066764174070192684977, −3.88356393907147498616077250815, −3.31263948021534646629842438332, −3.19350881248401727017421363242, −1.84443473513052294899388421909, −1.33760290872410049724095802904, −0.55372918428696028243036519220, 0.55372918428696028243036519220, 1.33760290872410049724095802904, 1.84443473513052294899388421909, 3.19350881248401727017421363242, 3.31263948021534646629842438332, 3.88356393907147498616077250815, 4.36370824066764174070192684977, 4.83491829718995064412919922657, 4.84266368455744552217499572239, 6.06649326926350063125449392390, 6.18918864564920868474958882471, 6.73075461733201898288233715476, 6.91176462577580700595313155801, 7.64794508068527693644007941078, 7.70415298038578573845057256038, 8.287773417794300723168477318689, 8.788506542317774987935002109008, 8.948549508496821513849176904799, 9.588603199068747606250465893550, 10.19047343942992710068432091141

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