Properties

Label 4-21e2-1.1-c16e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $1162.00$
Root an. cond. $5.83850$
Motivic weight $16$
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  − 6.56e3·3-s − 6.55e4·4-s − 1.14e7·7-s + 4.29e8·12-s − 2.41e9·13-s + 2.38e10·19-s + 7.52e10·21-s − 1.52e11·25-s + 2.82e11·27-s + 7.51e11·28-s + 2.02e11·31-s − 1.76e12·37-s + 1.58e13·39-s + 4.67e13·43-s + 9.83e13·49-s + 1.57e14·52-s − 1.56e14·57-s − 1.84e14·61-s + 2.81e14·64-s − 5.78e14·67-s − 1.43e15·73-s + 1.00e15·75-s − 1.56e15·76-s − 2.57e15·79-s − 1.85e15·81-s − 4.93e15·84-s + 2.76e16·91-s + ⋯
L(s)  = 1  − 3-s − 4-s − 1.99·7-s + 12-s − 2.95·13-s + 1.40·19-s + 1.99·21-s − 25-s + 27-s + 1.99·28-s + 0.237·31-s − 0.502·37-s + 2.95·39-s + 3.99·43-s + 2.96·49-s + 2.95·52-s − 1.40·57-s − 0.961·61-s + 64-s − 1.42·67-s − 1.78·73-s + 75-s − 1.40·76-s − 1.69·79-s − 81-s − 1.99·84-s + 5.88·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+8)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1162.00\)
Root analytic conductor: \(5.83850\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :8, 8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.09745625192\)
\(L(\frac12)\) \(\approx\) \(0.09745625192\)
\(L(9)\) 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$C_2$ \( 1 + p^{8} T + p^{16} T^{2} \)
7$C_2$ \( 1 + 11472481 T + p^{16} T^{2} \)
good2$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
5$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
11$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
13$C_2$ \( ( 1 + 1205410561 T + p^{16} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
19$C_2$ \( ( 1 - 32868566594 T + p^{16} T^{2} )( 1 + 9013552993 T + p^{16} T^{2} ) \)
23$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
31$C_2$ \( ( 1 - 1568167997954 T + p^{16} T^{2} )( 1 + 1365381008161 T + p^{16} T^{2} ) \)
37$C_2$ \( ( 1 - 5005401539519 T + p^{16} T^{2} )( 1 + 6771424503358 T + p^{16} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
43$C_2$ \( ( 1 - 23369166652127 T + p^{16} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
53$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
59$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
61$C_2$ \( ( 1 - 199031420303999 T + p^{16} T^{2} )( 1 + 383320135538113 T + p^{16} T^{2} ) \)
67$C_2$ \( ( 1 - 203869623664607 T + p^{16} T^{2} )( 1 + 782743712096446 T + p^{16} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
73$C_2$ \( ( 1 + 86674146082561 T + p^{16} T^{2} )( 1 + 1351474503392638 T + p^{16} T^{2} ) \)
79$C_2$ \( ( 1 - 102579982895519 T + p^{16} T^{2} )( 1 + 2677497415399678 T + p^{16} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{8} T )^{2}( 1 + p^{8} T )^{2} \)
89$C_2$ \( ( 1 - p^{8} T + p^{16} T^{2} )( 1 + p^{8} T + p^{16} T^{2} ) \)
97$C_2$ \( ( 1 - 15621585304991234 T + p^{16} T^{2} )^{2} \)
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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

−14.33456021006174895443124797327, −14.20870593026171018778816415169, −13.20280165745625455378174840976, −12.72050149621963080405138880711, −12.02861362738924948912521611023, −11.83676886597536103284025073196, −10.44475188677537248769035349558, −10.03118491712803341510074065792, −9.259947904168653561591912381712, −9.187543426116946574710320880828, −7.40559094551865126543128498029, −7.31807551412554410797416230302, −6.11375274074561265591952069944, −5.60520130029096884081248828227, −4.82053383880090644487836296140, −4.13967408207689331256391446081, −2.97968316825045336328788213138, −2.49034723332367872108629798395, −0.77192071892843547020338994618, −0.14919050651065554542519296887, 0.14919050651065554542519296887, 0.77192071892843547020338994618, 2.49034723332367872108629798395, 2.97968316825045336328788213138, 4.13967408207689331256391446081, 4.82053383880090644487836296140, 5.60520130029096884081248828227, 6.11375274074561265591952069944, 7.31807551412554410797416230302, 7.40559094551865126543128498029, 9.187543426116946574710320880828, 9.259947904168653561591912381712, 10.03118491712803341510074065792, 10.44475188677537248769035349558, 11.83676886597536103284025073196, 12.02861362738924948912521611023, 12.72050149621963080405138880711, 13.20280165745625455378174840976, 14.20870593026171018778816415169, 14.33456021006174895443124797327

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