Properties

Label 4-21e2-1.1-c13e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $507.082$
Root an. cond. $4.74536$
Motivic weight $13$
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  − 2.18e3·3-s − 8.19e3·4-s + 3.86e5·7-s + 3.18e6·9-s + 1.79e7·12-s − 6.40e8·19-s − 8.45e8·21-s + 1.22e9·25-s − 3.48e9·27-s − 3.16e9·28-s − 1.42e10·31-s − 2.61e10·36-s − 2.73e10·37-s + 1.54e11·43-s + 5.25e10·49-s + 1.40e12·57-s − 4.24e11·61-s + 1.23e12·63-s + 5.49e11·64-s + 2.26e11·67-s − 3.13e12·73-s − 2.66e12·75-s + 5.24e12·76-s − 4.08e12·79-s + 2.54e12·81-s + 6.92e12·84-s + 3.11e13·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s + 1.24·7-s + 2·9-s + 1.73·12-s − 3.12·19-s − 2.15·21-s + 25-s − 1.73·27-s − 1.24·28-s − 2.88·31-s − 2·36-s − 1.75·37-s + 3.72·43-s + 0.542·49-s + 5.40·57-s − 1.05·61-s + 2.48·63-s + 64-s + 0.306·67-s − 2.42·73-s − 1.73·75-s + 3.12·76-s − 1.89·79-s + 81-s + 2.15·84-s + 4.99·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(507.082\)
Root analytic conductor: \(4.74536\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :13/2, 13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(0.02177958999\)
\(L(\frac12)\) \(\approx\) \(0.02177958999\)
\(L(\frac{15}{2})\) 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$C_2$ \( 1 + p^{7} T + p^{13} T^{2} \)
7$C_2$ \( 1 - 386569 T + p^{13} T^{2} \)
good2$C_2^2$ \( 1 + p^{13} T^{2} + p^{26} T^{4} \)
5$C_2^2$ \( 1 - p^{13} T^{2} + p^{26} T^{4} \)
11$C_2^2$ \( 1 + p^{13} T^{2} + p^{26} T^{4} \)
13$C_2$ \( ( 1 - 34804217 T + p^{13} T^{2} )( 1 + 34804217 T + p^{13} T^{2} ) \)
17$C_2^2$ \( 1 - p^{13} T^{2} + p^{26} T^{4} \)
19$C_2$ \( ( 1 + 231547688 T + p^{13} T^{2} )( 1 + 408943081 T + p^{13} T^{2} ) \)
23$C_2^2$ \( 1 + p^{13} T^{2} + p^{26} T^{4} \)
29$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4383058669 T + p^{13} T^{2} )( 1 + 9862529036 T + p^{13} T^{2} ) \)
37$C_2$ \( ( 1 + 705453830 T + p^{13} T^{2} )( 1 + 26671973339 T + p^{13} T^{2} ) \)
41$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 77218825315 T + p^{13} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{13} T^{2} + p^{26} T^{4} \)
53$C_2^2$ \( 1 + p^{13} T^{2} + p^{26} T^{4} \)
59$C_2^2$ \( 1 - p^{13} T^{2} + p^{26} T^{4} \)
61$C_2$ \( ( 1 - 171215488093 T + p^{13} T^{2} )( 1 + 595391378401 T + p^{13} T^{2} ) \)
67$C_2$ \( ( 1 - 1380765177776 T + p^{13} T^{2} )( 1 + 1153893260515 T + p^{13} T^{2} ) \)
71$C_2$ \( ( 1 - p^{13} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 641895299153 T + p^{13} T^{2} )( 1 + 2490400654570 T + p^{13} T^{2} ) \)
79$C_2$ \( ( 1 + 820761767284 T + p^{13} T^{2} )( 1 + 3263778980263 T + p^{13} T^{2} ) \)
83$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{13} T^{2} + p^{26} T^{4} \)
97$C_2$ \( ( 1 - 12352800515314 T + p^{13} T^{2} )( 1 + 12352800515314 T + p^{13} T^{2} ) \)
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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

−15.42703791873672706935972752902, −14.43810863136158642230038571470, −14.41144220168637636957155604700, −12.95506579499731654026978607623, −12.84366572475378283387522069578, −12.15899771297532992789137558542, −11.12728922960652128071852931784, −10.82051557937845065578548123328, −10.42985297835329034365078192493, −8.978027651241435389342825827565, −8.794223336235236041743570603275, −7.56565459827740863164990487562, −6.79175274518170245486948119696, −5.83200029348347527848028710738, −5.25577649334413890177775018518, −4.33192393315346704332217222780, −4.20486654244552306664271556451, −2.10858437302157878704394914459, −1.28665748441672766305741244973, −0.06267998143854701620557538295, 0.06267998143854701620557538295, 1.28665748441672766305741244973, 2.10858437302157878704394914459, 4.20486654244552306664271556451, 4.33192393315346704332217222780, 5.25577649334413890177775018518, 5.83200029348347527848028710738, 6.79175274518170245486948119696, 7.56565459827740863164990487562, 8.794223336235236041743570603275, 8.978027651241435389342825827565, 10.42985297835329034365078192493, 10.82051557937845065578548123328, 11.12728922960652128071852931784, 12.15899771297532992789137558542, 12.84366572475378283387522069578, 12.95506579499731654026978607623, 14.41144220168637636957155604700, 14.43810863136158642230038571470, 15.42703791873672706935972752902

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