Properties

Label 4-21e4-1.1-c1e2-0-0
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
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 − 4·5-s − 3·16-s − 12·17-s + 4·20-s + 6·25-s − 4·37-s − 4·41-s + 8·43-s + 8·47-s − 8·59-s + 7·64-s − 8·67-s + 12·68-s + 12·80-s + 48·85-s − 4·89-s − 6·100-s + 4·101-s + 28·109-s − 10·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 3/4·16-s − 2.91·17-s + 0.894·20-s + 6/5·25-s − 0.657·37-s − 0.624·41-s + 1.21·43-s + 1.16·47-s − 1.04·59-s + 7/8·64-s − 0.977·67-s + 1.45·68-s + 1.34·80-s + 5.20·85-s − 0.423·89-s − 3/5·100-s + 0.398·101-s + 2.68·109-s − 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{194481} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3575884727\)
\(L(\frac12)\) \(\approx\) \(0.3575884727\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−9.018312456434419527119744159153, −8.693390433543831577647523795630, −8.251469284589790909670122189063, −7.64380613206652059735072783819, −7.26089867011393708858561091827, −6.79678312336036188663270632781, −6.35393046739437426289631565092, −5.62583091530747883907100729659, −4.73546190050037463375370585314, −4.49878458502574972078691617427, −4.11723676863830659898666685882, −3.56199012513494909227187173112, −2.69641876559375287050960973286, −1.97440148384713200215076620060, −0.36811147113713012302502489118, 0.36811147113713012302502489118, 1.97440148384713200215076620060, 2.69641876559375287050960973286, 3.56199012513494909227187173112, 4.11723676863830659898666685882, 4.49878458502574972078691617427, 4.73546190050037463375370585314, 5.62583091530747883907100729659, 6.35393046739437426289631565092, 6.79678312336036188663270632781, 7.26089867011393708858561091827, 7.64380613206652059735072783819, 8.251469284589790909670122189063, 8.693390433543831577647523795630, 9.018312456434419527119744159153

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