Properties

Label 4-103968-1.1-c1e2-0-4
Degree $4$
Conductor $103968$
Sign $1$
Analytic cond. $6.62908$
Root an. cond. $1.60458$
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  + 2-s + 4-s + 8-s + 9-s − 8·13-s + 16-s + 12·17-s + 18-s − 10·25-s − 8·26-s + 12·29-s + 32-s + 12·34-s + 36-s − 8·37-s + 12·41-s + 2·49-s − 10·50-s − 8·52-s + 12·53-s + 12·58-s + 28·61-s + 64-s + 12·68-s + 72-s + 28·73-s − 8·74-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s − 2.21·13-s + 1/4·16-s + 2.91·17-s + 0.235·18-s − 2·25-s − 1.56·26-s + 2.22·29-s + 0.176·32-s + 2.05·34-s + 1/6·36-s − 1.31·37-s + 1.87·41-s + 2/7·49-s − 1.41·50-s − 1.10·52-s + 1.64·53-s + 1.57·58-s + 3.58·61-s + 1/8·64-s + 1.45·68-s + 0.117·72-s + 3.27·73-s − 0.929·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(103968\)    =    \(2^{5} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(6.62908\)
Root analytic conductor: \(1.60458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{103968} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 103968,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.429312260\)
\(L(\frac12)\) \(\approx\) \(2.429312260\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−9.659160529040683538070775225454, −9.359231481850041434452307896895, −8.211124874872632719716882271254, −7.973335490421157373200340268575, −7.63111837112808582138120144993, −6.84732233829542762166868447522, −6.75095657741783063390476290272, −5.61719283222003359694793802526, −5.38776924769841264609355327167, −5.07485672472001477729499979724, −3.94709989510610133111766841641, −3.89674503164637365373404147869, −2.72251484544452322835217288408, −2.40790679424770113535591807077, −1.09773072794770812951577890272, 1.09773072794770812951577890272, 2.40790679424770113535591807077, 2.72251484544452322835217288408, 3.89674503164637365373404147869, 3.94709989510610133111766841641, 5.07485672472001477729499979724, 5.38776924769841264609355327167, 5.61719283222003359694793802526, 6.75095657741783063390476290272, 6.84732233829542762166868447522, 7.63111837112808582138120144993, 7.973335490421157373200340268575, 8.211124874872632719716882271254, 9.359231481850041434452307896895, 9.659160529040683538070775225454

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