Properties

Label 4-442e2-1.1-c1e2-0-14
Degree $4$
Conductor $195364$
Sign $1$
Analytic cond. $12.4565$
Root an. cond. $1.87866$
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  + 4-s + 2·9-s + 16-s − 2·19-s + 6·25-s + 2·36-s + 18·43-s + 6·47-s − 6·49-s − 12·59-s + 64-s − 8·67-s − 2·76-s − 5·81-s + 18·83-s + 24·89-s + 6·100-s − 6·101-s − 18·103-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s + 2/3·9-s + 1/4·16-s − 0.458·19-s + 6/5·25-s + 1/3·36-s + 2.74·43-s + 0.875·47-s − 6/7·49-s − 1.56·59-s + 1/8·64-s − 0.977·67-s − 0.229·76-s − 5/9·81-s + 1.97·83-s + 2.54·89-s + 3/5·100-s − 0.597·101-s − 1.77·103-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(195364\)    =    \(2^{2} \cdot 13^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(12.4565\)
Root analytic conductor: \(1.87866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{195364} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 195364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.068464020\)
\(L(\frac12)\) \(\approx\) \(2.068464020\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + p T^{2} \)
17$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \)
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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

−9.160025348108921351005727814663, −8.756942805022124848298409715946, −7.986813940716329283809796267027, −7.65587088668776506942585074128, −7.24811733169512396588291526889, −6.65966732532351866812391117022, −6.26370099721229970828023941604, −5.76301756388933689619530604886, −5.08180300610355499200586820841, −4.50013557354371053098837084146, −4.05890927806236970190699935434, −3.25730355379818314650391441499, −2.64626039628269555711061537078, −1.91085072921640392396861172153, −0.970818847478859526690702316139, 0.970818847478859526690702316139, 1.91085072921640392396861172153, 2.64626039628269555711061537078, 3.25730355379818314650391441499, 4.05890927806236970190699935434, 4.50013557354371053098837084146, 5.08180300610355499200586820841, 5.76301756388933689619530604886, 6.26370099721229970828023941604, 6.65966732532351866812391117022, 7.24811733169512396588291526889, 7.65587088668776506942585074128, 7.986813940716329283809796267027, 8.756942805022124848298409715946, 9.160025348108921351005727814663

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