Properties

Label 4-21e2-1.1-c15e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $897.939$
Root an. cond. $5.47408$
Motivic weight $15$
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  + 6.55e4·4-s − 1.24e6·7-s − 1.43e7·9-s + 3.22e9·16-s − 6.10e10·25-s − 8.15e10·28-s − 9.40e11·36-s − 2.18e12·37-s − 2.88e12·43-s − 3.19e12·49-s + 1.78e13·63-s + 1.40e14·64-s + 1.98e14·67-s − 1.77e14·79-s + 2.05e14·81-s − 4.00e15·100-s + 7.56e15·109-s − 4.01e15·112-s + 8.35e15·121-s + 127-s + 131-s + 137-s + 139-s − 4.62e16·144-s − 1.42e17·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s − 0.571·7-s − 9-s + 3·16-s − 2·25-s − 1.14·28-s − 2·36-s − 3.77·37-s − 1.61·43-s − 0.673·49-s + 0.571·63-s + 4·64-s + 3.99·67-s − 1.03·79-s + 81-s − 4·100-s + 3.96·109-s − 1.71·112-s + 2·121-s − 3·144-s − 7.55·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(897.939\)
Root analytic conductor: \(5.47408\)
Motivic weight: \(15\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :15/2, 15/2),\ 1)\)

Particular Values

\(L(8)\) \(\approx\) \(2.749166253\)
\(L(\frac12)\) \(\approx\) \(2.749166253\)
\(L(\frac{17}{2})\) 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$C_2$ \( 1 + p^{15} T^{2} \)
7$C_2$ \( 1 + 1244900 T + p^{15} T^{2} \)
good2$C_2$ \( ( 1 - p^{15} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{15} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 397771850 T + p^{15} T^{2} )( 1 + 397771850 T + p^{15} T^{2} ) \)
17$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7700827736 T + p^{15} T^{2} )( 1 + 7700827736 T + p^{15} T^{2} ) \)
23$C_2$ \( ( 1 - p^{15} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{15} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 213681227452 T + p^{15} T^{2} )( 1 + 213681227452 T + p^{15} T^{2} ) \)
37$C_2$ \( ( 1 + 1090158909950 T + p^{15} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 1440654152600 T + p^{15} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{15} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 40241378988902 T + p^{15} T^{2} )( 1 + 40241378988902 T + p^{15} T^{2} ) \)
67$C_2$ \( ( 1 - 99059017336400 T + p^{15} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{15} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 9014812804550 T + p^{15} T^{2} )( 1 + 9014812804550 T + p^{15} T^{2} ) \)
79$C_2$ \( ( 1 + 88692309079036 T + p^{15} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1035097921427150 T + p^{15} T^{2} )( 1 + 1035097921427150 T + p^{15} T^{2} ) \)
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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

−15.49324509119703693504009919440, −14.24082859765475427127382918343, −13.82914797863283444878026347711, −12.68591442736919373644606412592, −12.18309682557366742813087522021, −11.47215887844121575066050454796, −11.24357202339979074206706482467, −10.22921667753242064975918382585, −9.895183044066066667924107343752, −8.594137534394891621487304573129, −8.005847599001367050501114729258, −7.06305225098507092317833600335, −6.63712617479463320711074826739, −5.84155133974696674709532997182, −5.28425699911513668261171264844, −3.42968457279867998317305634211, −3.33454189204525733152853180207, −2.07261344498227344383184156871, −1.84438923898225316045784140169, −0.46919766004656336527176456339, 0.46919766004656336527176456339, 1.84438923898225316045784140169, 2.07261344498227344383184156871, 3.33454189204525733152853180207, 3.42968457279867998317305634211, 5.28425699911513668261171264844, 5.84155133974696674709532997182, 6.63712617479463320711074826739, 7.06305225098507092317833600335, 8.005847599001367050501114729258, 8.594137534394891621487304573129, 9.895183044066066667924107343752, 10.22921667753242064975918382585, 11.24357202339979074206706482467, 11.47215887844121575066050454796, 12.18309682557366742813087522021, 12.68591442736919373644606412592, 13.82914797863283444878026347711, 14.24082859765475427127382918343, 15.49324509119703693504009919440

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