Properties

Label 4-21e2-1.1-c9e2-0-1
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $116.980$
Root an. cond. $3.28873$
Motivic weight $9$
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  + 1.02e3·4-s + 1.25e4·7-s − 1.96e4·9-s + 7.86e5·16-s − 3.90e6·25-s + 1.28e7·28-s − 2.01e7·36-s + 3.07e7·37-s + 3.31e7·43-s + 1.17e8·49-s − 2.47e8·63-s + 5.36e8·64-s + 2.25e8·67-s − 1.23e9·79-s + 3.87e8·81-s − 4.00e9·100-s − 4.48e9·109-s + 9.89e9·112-s + 4.71e9·121-s + 127-s + 131-s + 137-s + 139-s − 1.54e10·144-s + 3.15e10·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s + 1.98·7-s − 9-s + 3·16-s − 2·25-s + 3.96·28-s − 2·36-s + 2.69·37-s + 1.47·43-s + 2.92·49-s − 1.98·63-s + 4·64-s + 1.36·67-s − 3.56·79-s + 81-s − 4·100-s − 3.04·109-s + 5.94·112-s + 2·121-s − 3·144-s + 5.39·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(116.980\)
Root analytic conductor: \(3.28873\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(5.028244090\)
\(L(\frac12)\) \(\approx\) \(5.028244090\)
\(L(\frac{11}{2})\) 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$C_2$ \( 1 + p^{9} T^{2} \)
7$C_2$ \( 1 - 12580 T + p^{9} T^{2} \)
good2$C_2$ \( ( 1 - p^{9} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{9} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 118370 T + p^{9} T^{2} )( 1 + 118370 T + p^{9} T^{2} ) \)
17$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 976696 T + p^{9} T^{2} )( 1 + 976696 T + p^{9} T^{2} ) \)
23$C_2$ \( ( 1 - p^{9} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{9} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 1691228 T + p^{9} T^{2} )( 1 + 1691228 T + p^{9} T^{2} ) \)
37$C_2$ \( ( 1 - 15384490 T + p^{9} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 16577080 T + p^{9} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{9} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 117903058 T + p^{9} T^{2} )( 1 + 117903058 T + p^{9} T^{2} ) \)
67$C_2$ \( ( 1 - 112542320 T + p^{9} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{9} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 296368310 T + p^{9} T^{2} )( 1 + 296368310 T + p^{9} T^{2} ) \)
79$C_2$ \( ( 1 + 616732324 T + p^{9} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{9} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 1288928270 T + p^{9} T^{2} )( 1 + 1288928270 T + p^{9} T^{2} ) \)
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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

−16.29347392102763095843923857324, −15.63754777203022527958331762027, −15.10484300392934121866015946783, −14.46199329357165201518707053773, −14.18843970965201350425561111340, −12.97877571823745448741738640611, −11.80874462466618219450408468180, −11.73809606064407243243495251782, −11.11650216966029399160245113548, −10.69070388322126610115725421790, −9.603965407851349776948850704874, −8.225825820609202210742760998110, −7.910473518399592841425495380416, −7.20426595214389768556104322562, −5.96265580908571004496031667237, −5.60535303463255262468288552571, −4.16070583168998177499642448006, −2.68411536862315119314997138525, −2.05998369827951405199695870376, −1.10418802412876644429856344345, 1.10418802412876644429856344345, 2.05998369827951405199695870376, 2.68411536862315119314997138525, 4.16070583168998177499642448006, 5.60535303463255262468288552571, 5.96265580908571004496031667237, 7.20426595214389768556104322562, 7.910473518399592841425495380416, 8.225825820609202210742760998110, 9.603965407851349776948850704874, 10.69070388322126610115725421790, 11.11650216966029399160245113548, 11.73809606064407243243495251782, 11.80874462466618219450408468180, 12.97877571823745448741738640611, 14.18843970965201350425561111340, 14.46199329357165201518707053773, 15.10484300392934121866015946783, 15.63754777203022527958331762027, 16.29347392102763095843923857324

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