Properties

Label 4-21e2-1.1-c26e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $8089.47$
Root an. cond. $9.48374$
Motivic weight $26$
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  + 1.59e6·3-s − 6.71e7·4-s − 4.43e10·7-s − 1.06e14·12-s + 1.21e15·13-s + 5.26e16·19-s − 7.06e16·21-s − 1.49e18·25-s − 4.05e18·27-s + 2.97e18·28-s + 1.88e19·31-s − 2.62e20·37-s + 1.93e21·39-s + 5.05e21·43-s − 7.42e21·49-s − 8.12e22·52-s + 8.39e22·57-s − 2.63e23·61-s + 3.02e23·64-s + 1.04e24·67-s − 7.32e22·73-s − 2.37e24·75-s − 3.53e24·76-s − 7.34e24·79-s − 6.46e24·81-s + 4.74e24·84-s − 5.37e25·91-s + ⋯
L(s)  = 1  + 3-s − 4-s − 0.457·7-s − 12-s + 3.99·13-s + 1.25·19-s − 0.457·21-s − 25-s − 27-s + 0.457·28-s + 0.770·31-s − 1.07·37-s + 3.99·39-s + 2.94·43-s − 0.790·49-s − 3.99·52-s + 1.25·57-s − 1.62·61-s + 64-s + 1.90·67-s − 0.0438·73-s − 75-s − 1.25·76-s − 1.57·79-s − 81-s + 0.457·84-s − 1.83·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+13)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(8089.47\)
Root analytic conductor: \(9.48374\)
Motivic weight: \(26\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :13, 13),\ 1)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(3.743950654\)
\(L(\frac12)\) \(\approx\) \(3.743950654\)
\(L(14)\) 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$C_2$ \( 1 - p^{13} T + p^{26} T^{2} \)
7$C_2$ \( 1 + 44342429053 T + p^{26} T^{2} \)
good2$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
5$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
11$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
13$C_2$ \( ( 1 - 605583307798583 T + p^{26} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
19$C_2$ \( ( 1 - 83128476573258443 T + p^{26} T^{2} )( 1 + 30491635106368774 T + p^{26} T^{2} ) \)
23$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
31$C_2$ \( ( 1 - 48434386391053004114 T + p^{26} T^{2} )( 1 + 29623889298994033621 T + p^{26} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(22\!\cdots\!27\)\( T + p^{26} T^{2} )( 1 + \)\(48\!\cdots\!94\)\( T + p^{26} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
43$C_2$ \( ( 1 - \)\(25\!\cdots\!39\)\( T + p^{26} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
53$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
59$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
61$C_2$ \( ( 1 - \)\(30\!\cdots\!39\)\( T + p^{26} T^{2} )( 1 + \)\(29\!\cdots\!13\)\( T + p^{26} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(81\!\cdots\!02\)\( T + p^{26} T^{2} )( 1 - \)\(23\!\cdots\!51\)\( T + p^{26} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
73$C_2$ \( ( 1 - \)\(28\!\cdots\!34\)\( T + p^{26} T^{2} )( 1 + \)\(29\!\cdots\!57\)\( T + p^{26} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(13\!\cdots\!91\)\( T + p^{26} T^{2} )( 1 + \)\(86\!\cdots\!22\)\( T + p^{26} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{13} T )^{2}( 1 + p^{13} T )^{2} \)
89$C_2$ \( ( 1 - p^{13} T + p^{26} T^{2} )( 1 + p^{13} T + p^{26} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(17\!\cdots\!42\)\( T + p^{26} T^{2} )^{2} \)
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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

−13.32381047089597132606504649904, −12.50599051469148240683152600299, −11.40617721508372390948455261309, −11.14364620105895236218213046569, −10.30339532220748051934524850015, −9.369025326527607583342713417116, −9.125765715929274091943971015773, −8.494770220855404571050823665338, −8.205187818637520232837075097595, −7.38692852373053955799116816392, −6.20765730309131052170529698194, −6.03646802104080837047462137569, −5.21838446823022013211669108582, −3.96720371620380582513693058182, −3.91599865559192102179840867002, −3.31418693964327077438179955223, −2.64138877333705113488400476643, −1.55562760658935783979806868618, −1.13851501663217743589977387889, −0.46144697519819090620871322825, 0.46144697519819090620871322825, 1.13851501663217743589977387889, 1.55562760658935783979806868618, 2.64138877333705113488400476643, 3.31418693964327077438179955223, 3.91599865559192102179840867002, 3.96720371620380582513693058182, 5.21838446823022013211669108582, 6.03646802104080837047462137569, 6.20765730309131052170529698194, 7.38692852373053955799116816392, 8.205187818637520232837075097595, 8.494770220855404571050823665338, 9.125765715929274091943971015773, 9.369025326527607583342713417116, 10.30339532220748051934524850015, 11.14364620105895236218213046569, 11.40617721508372390948455261309, 12.50599051469148240683152600299, 13.32381047089597132606504649904

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