Properties

Label 4-21e2-1.1-c3e2-0-2
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $1.53522$
Root an. cond. $1.11312$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s − 3·3-s + 8·4-s + 3·5-s − 9·6-s − 7·7-s + 45·8-s + 9·10-s + 15·11-s − 24·12-s − 128·13-s − 21·14-s − 9·15-s + 135·16-s − 84·17-s + 16·19-s + 24·20-s + 21·21-s + 45·22-s + 84·23-s − 135·24-s + 125·25-s − 384·26-s + 27·27-s − 56·28-s − 594·29-s − 27·30-s + ⋯
L(s)  = 1  + 1.06·2-s − 0.577·3-s + 4-s + 0.268·5-s − 0.612·6-s − 0.377·7-s + 1.98·8-s + 0.284·10-s + 0.411·11-s − 0.577·12-s − 2.73·13-s − 0.400·14-s − 0.154·15-s + 2.10·16-s − 1.19·17-s + 0.193·19-s + 0.268·20-s + 0.218·21-s + 0.436·22-s + 0.761·23-s − 1.14·24-s + 25-s − 2.89·26-s + 0.192·27-s − 0.377·28-s − 3.80·29-s − 0.164·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.53522\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.770138298\)
\(L(\frac12)\) \(\approx\) \(1.770138298\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p T + p^{2} T^{2} \)
7$C_2$ \( 1 + p T + p^{3} T^{2} \)
good2$C_2^2$ \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 3 T - 116 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 64 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - 16 T - 6603 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 297 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 253 T + 34218 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 360 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 26 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T - 102923 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 363 T - 17108 T^{2} + 363 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 15 T - 205154 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 118 T - 213057 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 342 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 362 T - 257973 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 477 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 906 T + 115867 T^{2} + 906 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 503 T + p^{3} T^{2} )^{2} \)
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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

−17.58386982955827734240686480395, −17.23074733756588066166355793207, −16.67477650348304783612437507985, −16.30416758363011446028043546281, −15.27106905125728587197844724311, −14.60107513759248829817160017074, −14.40025286648667157123636101921, −13.11785989273716789651633747066, −13.00932163019732403024923293547, −12.28229638658006112017284905068, −11.20719166378544860614006310948, −11.07506946322079037619428719899, −9.873668919387526237337095414518, −9.361730012047625925671515567963, −7.51666512192279009581180184974, −7.24478076920553501392747320947, −6.10912565339455725894963758727, −5.03511410905422182392571092911, −4.37886213680069290113882929938, −2.45162046213851339139222966109, 2.45162046213851339139222966109, 4.37886213680069290113882929938, 5.03511410905422182392571092911, 6.10912565339455725894963758727, 7.24478076920553501392747320947, 7.51666512192279009581180184974, 9.361730012047625925671515567963, 9.873668919387526237337095414518, 11.07506946322079037619428719899, 11.20719166378544860614006310948, 12.28229638658006112017284905068, 13.00932163019732403024923293547, 13.11785989273716789651633747066, 14.40025286648667157123636101921, 14.60107513759248829817160017074, 15.27106905125728587197844724311, 16.30416758363011446028043546281, 16.67477650348304783612437507985, 17.23074733756588066166355793207, 17.58386982955827734240686480395

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