Properties

Label 4-332928-1.1-c1e2-0-16
Degree $4$
Conductor $332928$
Sign $1$
Analytic cond. $21.2277$
Root an. cond. $2.14647$
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 + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s − 8·11-s + 2·12-s + 16-s + 2·17-s + 3·18-s + 8·19-s − 8·22-s + 2·24-s − 6·25-s + 4·27-s + 32-s − 16·33-s + 2·34-s + 3·36-s + 8·38-s + 20·41-s + 24·43-s − 8·44-s + 2·48-s − 14·49-s − 6·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 9-s − 2.41·11-s + 0.577·12-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 1.83·19-s − 1.70·22-s + 0.408·24-s − 6/5·25-s + 0.769·27-s + 0.176·32-s − 2.78·33-s + 0.342·34-s + 1/2·36-s + 1.29·38-s + 3.12·41-s + 3.65·43-s − 1.20·44-s + 0.288·48-s − 2·49-s − 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(332928\)    =    \(2^{7} \cdot 3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(21.2277\)
Root analytic conductor: \(2.14647\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{332928} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 332928,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.157881714\)
\(L(\frac12)\) \(\approx\) \(4.157881714\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.658113684153939465660001552715, −7.958882066985845549410362974675, −7.890763986910657579747142309970, −7.36440720073199929079936076774, −7.34292617858622580122915984824, −6.29236664701447308952901966222, −5.62906231402718523510730620943, −5.51457882926074302977186886250, −4.91746541938042376263088347369, −4.13875090005906653574798019221, −3.84610425166553964062381179508, −2.86767294411575254972359726206, −2.76309813409348720224135008444, −2.21434037861733189800471556077, −1.01738437769258744176787321955, 1.01738437769258744176787321955, 2.21434037861733189800471556077, 2.76309813409348720224135008444, 2.86767294411575254972359726206, 3.84610425166553964062381179508, 4.13875090005906653574798019221, 4.91746541938042376263088347369, 5.51457882926074302977186886250, 5.62906231402718523510730620943, 6.29236664701447308952901966222, 7.34292617858622580122915984824, 7.36440720073199929079936076774, 7.890763986910657579747142309970, 7.958882066985845549410362974675, 8.658113684153939465660001552715

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