Properties

Label 4-21e2-1.1-c17e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $1480.45$
Root an. cond. $6.20295$
Motivic weight $17$
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  + 1.96e4·3-s − 1.31e5·4-s − 2.75e6·7-s + 2.58e8·9-s − 2.57e9·12-s − 2.53e11·19-s − 5.42e10·21-s + 7.62e11·25-s + 2.54e12·27-s + 3.61e11·28-s − 8.74e12·31-s − 3.38e13·36-s − 4.21e13·37-s − 1.09e14·43-s − 2.25e14·49-s − 4.99e15·57-s − 5.18e15·61-s − 7.12e14·63-s + 2.25e15·64-s + 2.69e14·67-s + 1.49e16·73-s + 1.50e16·75-s + 3.32e16·76-s + 2.64e16·79-s + 1.66e16·81-s + 7.11e15·84-s − 1.72e17·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s − 0.180·7-s + 2·9-s − 1.73·12-s − 3.43·19-s − 0.313·21-s + 25-s + 1.73·27-s + 0.180·28-s − 1.84·31-s − 2·36-s − 1.97·37-s − 1.43·43-s − 0.967·49-s − 5.94·57-s − 3.46·61-s − 0.361·63-s + 64-s + 0.0810·67-s + 2.17·73-s + 1.73·75-s + 3.43·76-s + 1.96·79-s + 81-s + 0.313·84-s − 3.19·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1480.45\)
Root analytic conductor: \(6.20295\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :17/2, 17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(0.7397357773\)
\(L(\frac12)\) \(\approx\) \(0.7397357773\)
\(L(\frac{19}{2})\) 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$C_2$ \( 1 - p^{9} T + p^{17} T^{2} \)
7$C_2$ \( 1 + 2758181 T + p^{17} T^{2} \)
good2$C_2^2$ \( 1 + p^{17} T^{2} + p^{34} T^{4} \)
5$C_2^2$ \( 1 - p^{17} T^{2} + p^{34} T^{4} \)
11$C_2^2$ \( 1 + p^{17} T^{2} + p^{34} T^{4} \)
13$C_2$ \( ( 1 - 3266839877 T + p^{17} T^{2} )( 1 + 3266839877 T + p^{17} T^{2} ) \)
17$C_2^2$ \( 1 - p^{17} T^{2} + p^{34} T^{4} \)
19$C_2$ \( ( 1 + 116632805768 T + p^{17} T^{2} )( 1 + 137302173991 T + p^{17} T^{2} ) \)
23$C_2^2$ \( 1 + p^{17} T^{2} + p^{34} T^{4} \)
29$C_2$ \( ( 1 - p^{17} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 351446059196 T + p^{17} T^{2} )( 1 + 8395073038759 T + p^{17} T^{2} ) \)
37$C_2$ \( ( 1 + 14723537266559 T + p^{17} T^{2} )( 1 + 27378365365430 T + p^{17} T^{2} ) \)
41$C_2$ \( ( 1 + p^{17} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 54897704698835 T + p^{17} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{17} T^{2} + p^{34} T^{4} \)
53$C_2^2$ \( 1 + p^{17} T^{2} + p^{34} T^{4} \)
59$C_2^2$ \( 1 - p^{17} T^{2} + p^{34} T^{4} \)
61$C_2$ \( ( 1 + 2528437643672447 T + p^{17} T^{2} )( 1 + 2653755538627021 T + p^{17} T^{2} ) \)
67$C_2$ \( ( 1 - 5886945232004336 T + p^{17} T^{2} )( 1 + 5617606043215765 T + p^{17} T^{2} ) \)
71$C_2$ \( ( 1 - p^{17} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 12856663699194230 T + p^{17} T^{2} )( 1 - 2131541818135687 T + p^{17} T^{2} ) \)
79$C_2$ \( ( 1 - 17716682078842556 T + p^{17} T^{2} )( 1 - 8750552126467667 T + p^{17} T^{2} ) \)
83$C_2$ \( ( 1 + p^{17} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{17} T^{2} + p^{34} T^{4} \)
97$C_2$ \( ( 1 - 118297695891443726 T + p^{17} T^{2} )( 1 + 118297695891443726 T + p^{17} T^{2} ) \)
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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

−14.52442582796849976908594628050, −13.74031202303732162307301646737, −13.43189344608350551498973227791, −12.58127377865421462491688527886, −12.52906423313397648282384532445, −10.72987156030919159941267717933, −10.62002563956613514908130527719, −9.334461195804108735254837475357, −9.212154440367386356237567623329, −8.311901461894302737167807047891, −8.276911895899791172137384170713, −6.99244850594775641708476934736, −6.47156239331448620788643065646, −5.06743510926847890243346793662, −4.37742455486164328761981449108, −3.77792064457441427014083380539, −3.09718009268942359849182785808, −2.04859868907675581650995899585, −1.71783164558578473133297397705, −0.21030744653775516846655122657, 0.21030744653775516846655122657, 1.71783164558578473133297397705, 2.04859868907675581650995899585, 3.09718009268942359849182785808, 3.77792064457441427014083380539, 4.37742455486164328761981449108, 5.06743510926847890243346793662, 6.47156239331448620788643065646, 6.99244850594775641708476934736, 8.276911895899791172137384170713, 8.311901461894302737167807047891, 9.212154440367386356237567623329, 9.334461195804108735254837475357, 10.62002563956613514908130527719, 10.72987156030919159941267717933, 12.52906423313397648282384532445, 12.58127377865421462491688527886, 13.43189344608350551498973227791, 13.74031202303732162307301646737, 14.52442582796849976908594628050

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