Properties

Label 4-21e2-1.1-c34e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $23646.4$
Root an. cond. $12.4005$
Motivic weight $34$
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  + 1.29e8·3-s − 1.71e10·4-s − 4.57e14·7-s − 2.21e18·12-s − 1.32e19·13-s + 1.05e22·19-s − 5.91e22·21-s − 5.82e23·25-s − 2.15e24·27-s + 7.86e24·28-s − 1.95e25·31-s − 8.59e26·37-s − 1.71e27·39-s − 1.74e28·43-s + 1.55e29·49-s + 2.27e29·52-s + 1.36e30·57-s + 4.46e30·61-s + 5.07e30·64-s + 2.20e31·67-s − 2.00e31·73-s − 7.51e31·75-s − 1.80e32·76-s − 3.36e32·79-s − 2.78e32·81-s + 1.01e33·84-s + 6.06e33·91-s + ⋯
L(s)  = 1  + 3-s − 4-s − 1.96·7-s − 12-s − 1.53·13-s + 1.92·19-s − 1.96·21-s − 25-s − 27-s + 1.96·28-s − 0.869·31-s − 1.88·37-s − 1.53·39-s − 2.97·43-s + 2.87·49-s + 1.53·52-s + 1.92·57-s + 1.99·61-s + 64-s + 1.99·67-s − 0.422·73-s − 75-s − 1.92·76-s − 1.85·79-s − 81-s + 1.96·84-s + 3.01·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+17)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(23646.4\)
Root analytic conductor: \(12.4005\)
Motivic weight: \(34\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :17, 17),\ 1)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(1.268189777\)
\(L(\frac12)\) \(\approx\) \(1.268189777\)
\(L(18)\) 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$C_2$ \( 1 - p^{17} T + p^{34} T^{2} \)
7$C_2$ \( 1 + 457653465545653 T + p^{34} T^{2} \)
good2$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
5$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
11$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
13$C_2$ \( ( 1 + 6628589056805300737 T + p^{34} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
19$C_2$ \( ( 1 - \)\(78\!\cdots\!03\)\( T + p^{34} T^{2} )( 1 - \)\(26\!\cdots\!46\)\( T + p^{34} T^{2} ) \)
23$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{17} T )^{2}( 1 + p^{17} T )^{2} \)
31$C_2$ \( ( 1 - \)\(25\!\cdots\!59\)\( T + p^{34} T^{2} )( 1 + \)\(44\!\cdots\!06\)\( T + p^{34} T^{2} ) \)
37$C_2$ \( ( 1 + \)\(16\!\cdots\!34\)\( T + p^{34} T^{2} )( 1 + \)\(69\!\cdots\!53\)\( T + p^{34} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{17} T )^{2}( 1 + p^{17} T )^{2} \)
43$C_2$ \( ( 1 + \)\(87\!\cdots\!61\)\( T + p^{34} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
53$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
59$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
61$C_2$ \( ( 1 - \)\(25\!\cdots\!99\)\( T + p^{34} T^{2} )( 1 - \)\(19\!\cdots\!67\)\( T + p^{34} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(12\!\cdots\!42\)\( T + p^{34} T^{2} )( 1 - \)\(94\!\cdots\!71\)\( T + p^{34} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{17} T )^{2}( 1 + p^{17} T )^{2} \)
73$C_2$ \( ( 1 - \)\(70\!\cdots\!94\)\( T + p^{34} T^{2} )( 1 + \)\(90\!\cdots\!37\)\( T + p^{34} T^{2} ) \)
79$C_2$ \( ( 1 + \)\(49\!\cdots\!82\)\( T + p^{34} T^{2} )( 1 + \)\(28\!\cdots\!29\)\( T + p^{34} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{17} T )^{2}( 1 + p^{17} T )^{2} \)
89$C_2$ \( ( 1 - p^{17} T + p^{34} T^{2} )( 1 + p^{17} T + p^{34} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(20\!\cdots\!02\)\( T + p^{34} T^{2} )^{2} \)
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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

−11.92892327972585138115068638132, −11.45767033506081402323746126972, −10.02988359020886324063871312727, −9.900583166499109113097063435412, −9.617214174830801738127082890776, −8.928591064303356451406515489443, −8.530978379986617246910474480663, −7.72139976650317004555121011774, −7.00918517389730965478980366061, −6.82503761052307169169953641636, −5.54450779468948681639259078644, −5.44134045562003713435175293875, −4.59353984637695093275728395201, −3.73863398151794588108003426572, −3.30481785399499313485791623070, −3.13697337204119849076936390629, −2.22978357247250504686401693917, −1.80771587328188239866442519094, −0.53133401046819750710553415806, −0.36810713432593345669034082371, 0.36810713432593345669034082371, 0.53133401046819750710553415806, 1.80771587328188239866442519094, 2.22978357247250504686401693917, 3.13697337204119849076936390629, 3.30481785399499313485791623070, 3.73863398151794588108003426572, 4.59353984637695093275728395201, 5.44134045562003713435175293875, 5.54450779468948681639259078644, 6.82503761052307169169953641636, 7.00918517389730965478980366061, 7.72139976650317004555121011774, 8.530978379986617246910474480663, 8.928591064303356451406515489443, 9.617214174830801738127082890776, 9.900583166499109113097063435412, 10.02988359020886324063871312727, 11.45767033506081402323746126972, 11.92892327972585138115068638132

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