Properties

Degree $4$
Conductor $998784$
Sign $1$
Motivic weight $1$
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 − 4·13-s − 4·15-s + 2·17-s + 2·25-s + 27-s + 12·29-s − 16·31-s − 4·39-s − 12·41-s − 4·45-s − 14·49-s + 2·51-s − 8·59-s + 16·65-s + 2·75-s + 16·79-s + 81-s + 8·83-s − 8·85-s + 12·87-s − 16·93-s + 36·113-s − 4·117-s − 6·121-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.10·13-s − 1.03·15-s + 0.485·17-s + 2/5·25-s + 0.192·27-s + 2.22·29-s − 2.87·31-s − 0.640·39-s − 1.87·41-s − 0.596·45-s − 2·49-s + 0.280·51-s − 1.04·59-s + 1.98·65-s + 0.230·75-s + 1.80·79-s + 1/9·81-s + 0.878·83-s − 0.867·85-s + 1.28·87-s − 1.65·93-s + 3.38·113-s − 0.369·117-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 998784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 998784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(998784\)    =    \(2^{7} \cdot 3^{3} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{998784} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 998784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7958503784\)
\(L(\frac12)\) \(\approx\) \(0.7958503784\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 - T \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.173282349834764667604075089133, −7.66498871915988097333678887917, −7.26958585488936859387169650645, −7.18155599764795540835879645469, −6.42703115208668412831343638258, −6.03138344128444123566991871629, −5.04277040000068891801244440280, −5.01357758335939619070346679818, −4.42764839672826892298257893061, −3.80218561106952661254790284491, −3.39638272741083957218437360098, −3.14826068230388029805668446658, −2.23113257513110131986739736948, −1.62857614603859772213021712939, −0.39064935037816856965515086908, 0.39064935037816856965515086908, 1.62857614603859772213021712939, 2.23113257513110131986739736948, 3.14826068230388029805668446658, 3.39638272741083957218437360098, 3.80218561106952661254790284491, 4.42764839672826892298257893061, 5.01357758335939619070346679818, 5.04277040000068891801244440280, 6.03138344128444123566991871629, 6.42703115208668412831343638258, 7.18155599764795540835879645469, 7.26958585488936859387169650645, 7.66498871915988097333678887917, 8.173282349834764667604075089133

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