Properties

Label 4-332928-1.1-c1e2-0-11
Degree $4$
Conductor $332928$
Sign $1$
Analytic cond. $21.2277$
Root an. cond. $2.14647$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 3·11-s + 12-s + 16-s + 2·17-s + 2·18-s + 13·19-s − 3·22-s − 24-s − 25-s − 5·27-s − 32-s + 3·33-s − 2·34-s − 2·36-s − 13·38-s − 9·41-s + 10·43-s + 3·44-s + 48-s + 2·49-s + 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 1/4·16-s + 0.485·17-s + 0.471·18-s + 2.98·19-s − 0.639·22-s − 0.204·24-s − 1/5·25-s − 0.962·27-s − 0.176·32-s + 0.522·33-s − 0.342·34-s − 1/3·36-s − 2.10·38-s − 1.40·41-s + 1.52·43-s + 0.452·44-s + 0.144·48-s + 2/7·49-s + 0.141·50-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 332928,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.742490761\)
\(L(\frac12)\) \(\approx\) \(1.742490761\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 104 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
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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

−8.811723905003011103122611844500, −8.299723172721116686868063741437, −7.902852090151395082219190081451, −7.51484488236082419263154063767, −6.97734156839502383447163376907, −6.67656822009036057836582092526, −5.78471426664507380287748963755, −5.54106422056457365540611603194, −5.11134632508443097815544755241, −4.09667539209628389133032262247, −3.61064246603099029136454278537, −3.07179752345030561285851433629, −2.57859676972452693967934789690, −1.60438834038346562981574728033, −0.883374070888861306142233225682, 0.883374070888861306142233225682, 1.60438834038346562981574728033, 2.57859676972452693967934789690, 3.07179752345030561285851433629, 3.61064246603099029136454278537, 4.09667539209628389133032262247, 5.11134632508443097815544755241, 5.54106422056457365540611603194, 5.78471426664507380287748963755, 6.67656822009036057836582092526, 6.97734156839502383447163376907, 7.51484488236082419263154063767, 7.902852090151395082219190081451, 8.299723172721116686868063741437, 8.811723905003011103122611844500

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