Properties

Label 4-21e4-1.1-c2e2-0-6
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $144.393$
Root an. cond. $3.46646$
Motivic weight $2$
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  − 8·4-s + 48·16-s + 50·25-s + 146·37-s + 122·43-s − 256·64-s − 26·67-s + 22·79-s − 400·100-s − 142·109-s − 242·121-s + 127-s + 131-s + 137-s + 139-s − 1.16e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 191·169-s − 976·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s + 2·25-s + 3.94·37-s + 2.83·43-s − 4·64-s − 0.388·67-s + 0.278·79-s − 4·100-s − 1.30·109-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.89·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.13·169-s − 5.67·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(144.393\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.493726169\)
\(L(\frac12)\) \(\approx\) \(1.493726169\)
\(L(2)\) 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$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
11$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
19$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \)
23$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 - 73 T + p^{2} T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2$ \( ( 1 - 61 T + p^{2} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 74 T + p^{2} T^{2} ) \)
67$C_2$ \( ( 1 + 13 T + p^{2} T^{2} )^{2} \)
71$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \)
79$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \)
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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

−10.89707641539154980596377826453, −10.73439648968899534058696743640, −10.15524653090223119618197443881, −9.540460978679954416689709443905, −9.231667596392539947081483964347, −9.178135075269650897683331185780, −8.312183899720989695300846506065, −8.191222601684563690203151960256, −7.57158796198353504656296830374, −7.14395071079933711743176236433, −6.10856293450598234469053637259, −6.02678892702023986165466494787, −5.25306688999071825885957542209, −4.80555258545325935908296396160, −4.16377038511814639966701223946, −4.12760894767927516786191289460, −3.06943119549509153015561609338, −2.60225555093138646316708067837, −1.09235090126459240591045842419, −0.67987782170374530752488418708, 0.67987782170374530752488418708, 1.09235090126459240591045842419, 2.60225555093138646316708067837, 3.06943119549509153015561609338, 4.12760894767927516786191289460, 4.16377038511814639966701223946, 4.80555258545325935908296396160, 5.25306688999071825885957542209, 6.02678892702023986165466494787, 6.10856293450598234469053637259, 7.14395071079933711743176236433, 7.57158796198353504656296830374, 8.191222601684563690203151960256, 8.312183899720989695300846506065, 9.178135075269650897683331185780, 9.231667596392539947081483964347, 9.540460978679954416689709443905, 10.15524653090223119618197443881, 10.73439648968899534058696743640, 10.89707641539154980596377826453

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