Properties

Degree 4
Conductor $ 2^{2} \cdot 19^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 6-s − 2·7-s − 8-s + 3·9-s − 12·11-s − 5·13-s − 2·14-s − 16-s − 3·17-s + 3·18-s + 2·21-s − 12·22-s − 3·23-s + 24-s + 5·25-s − 5·26-s − 8·27-s − 9·29-s − 8·31-s + 12·33-s − 3·34-s + 4·37-s + 5·39-s + 2·42-s − 8·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 9-s − 3.61·11-s − 1.38·13-s − 0.534·14-s − 1/4·16-s − 0.727·17-s + 0.707·18-s + 0.436·21-s − 2.55·22-s − 0.625·23-s + 0.204·24-s + 25-s − 0.980·26-s − 1.53·27-s − 1.67·29-s − 1.43·31-s + 2.08·33-s − 0.514·34-s + 0.657·37-s + 0.800·39-s + 0.308·42-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(521284\)    =    \(2^{2} \cdot 19^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{722} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 521284,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;19\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + T^{2} \)
19 \( 1 \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.25765001529881292714042737125, −9.886190533135730460502822307684, −9.446100495936444896176596385838, −9.169466048648994520446296461214, −8.092572216291284359554443608429, −8.056905948556871485554174373473, −7.49553318287091591647380974186, −7.12814032116340885760729957749, −6.72641471866208949390230551007, −5.94820683754930139030370328507, −5.53479944941460380950923957990, −5.23918886731926048487804639182, −4.79104111464750286928878710427, −4.46515944968277356981876872342, −3.51265975921653664699823417975, −3.17373830502963259637811322791, −2.20643893612872520195976331227, −2.19454637273018313789251851896, 0, 0, 2.19454637273018313789251851896, 2.20643893612872520195976331227, 3.17373830502963259637811322791, 3.51265975921653664699823417975, 4.46515944968277356981876872342, 4.79104111464750286928878710427, 5.23918886731926048487804639182, 5.53479944941460380950923957990, 5.94820683754930139030370328507, 6.72641471866208949390230551007, 7.12814032116340885760729957749, 7.49553318287091591647380974186, 8.056905948556871485554174373473, 8.092572216291284359554443608429, 9.169466048648994520446296461214, 9.446100495936444896176596385838, 9.886190533135730460502822307684, 10.25765001529881292714042737125

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