Properties

Label 4-21e4-1.1-c1e2-0-6
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
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  + 3·4-s + 5·16-s − 10·25-s + 12·37-s + 24·43-s + 3·64-s + 8·67-s + 16·79-s − 30·100-s − 36·109-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 36·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 72·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 3/2·4-s + 5/4·16-s − 2·25-s + 1.97·37-s + 3.65·43-s + 3/8·64-s + 0.977·67-s + 1.80·79-s − 3·100-s − 3.44·109-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.95·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 5.48·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-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: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.491796100\)
\(L(\frac12)\) \(\approx\) \(2.491796100\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{2} \)
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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

−11.17895832286995930823270705118, −10.87499971733367981585631075971, −10.82216115514045736533181030394, −9.957838963733577647122838300373, −9.535213828304778577885788498129, −9.350293495462748629510472403158, −8.520751961180749240367235472642, −7.87819863379023890748178489750, −7.72390490908741287354403524651, −7.24600568074370316077773975368, −6.70112406592555219308003279582, −6.02340198806629958935966074475, −5.97076295085884490558763280650, −5.34158610926627434786098273374, −4.36693795227291559219296206399, −4.00801398992300897065524229330, −3.21673288771012501952651827597, −2.38459610907231863518447469134, −2.20191138215244987282741431171, −1.04262763156758973621639725624, 1.04262763156758973621639725624, 2.20191138215244987282741431171, 2.38459610907231863518447469134, 3.21673288771012501952651827597, 4.00801398992300897065524229330, 4.36693795227291559219296206399, 5.34158610926627434786098273374, 5.97076295085884490558763280650, 6.02340198806629958935966074475, 6.70112406592555219308003279582, 7.24600568074370316077773975368, 7.72390490908741287354403524651, 7.87819863379023890748178489750, 8.520751961180749240367235472642, 9.350293495462748629510472403158, 9.535213828304778577885788498129, 9.957838963733577647122838300373, 10.82216115514045736533181030394, 10.87499971733367981585631075971, 11.17895832286995930823270705118

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