Properties

Label 4-189728-1.1-c1e2-0-8
Degree $4$
Conductor $189728$
Sign $1$
Analytic cond. $12.0972$
Root an. cond. $1.86496$
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 + 4-s + 8-s + 2·9-s + 16-s + 2·18-s + 10·25-s − 4·29-s + 32-s + 2·36-s − 4·37-s − 7·49-s + 10·50-s + 28·53-s − 4·58-s + 64-s + 2·72-s − 4·74-s − 5·81-s − 7·98-s + 10·100-s + 28·106-s − 4·109-s + 4·113-s − 4·116-s + 121-s + 127-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s + 1/4·16-s + 0.471·18-s + 2·25-s − 0.742·29-s + 0.176·32-s + 1/3·36-s − 0.657·37-s − 49-s + 1.41·50-s + 3.84·53-s − 0.525·58-s + 1/8·64-s + 0.235·72-s − 0.464·74-s − 5/9·81-s − 0.707·98-s + 100-s + 2.71·106-s − 0.383·109-s + 0.376·113-s − 0.371·116-s + 1/11·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(189728\)    =    \(2^{5} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(12.0972\)
Root analytic conductor: \(1.86496\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{189728} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 189728,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.892774765\)
\(L(\frac12)\) \(\approx\) \(2.892774765\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_2$ \( 1 + p T^{2} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.138355316340654241373080875063, −8.550966502145525084864492626185, −8.282024530209696904037410734491, −7.44090205001602193090431329744, −7.07845764333392661979449060345, −6.84693669503363575874611229048, −6.15586689609603382481791658368, −5.55424893264437314769560336816, −5.13419306901755155276020089858, −4.55978687407873991200208708116, −4.01880193632796957500043528001, −3.43901875101765770181944718211, −2.75386392283221271014365340678, −2.01705469370341431396968543751, −1.05865158925595700954317475021, 1.05865158925595700954317475021, 2.01705469370341431396968543751, 2.75386392283221271014365340678, 3.43901875101765770181944718211, 4.01880193632796957500043528001, 4.55978687407873991200208708116, 5.13419306901755155276020089858, 5.55424893264437314769560336816, 6.15586689609603382481791658368, 6.84693669503363575874611229048, 7.07845764333392661979449060345, 7.44090205001602193090431329744, 8.282024530209696904037410734491, 8.550966502145525084864492626185, 9.138355316340654241373080875063

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