Properties

Label 4-21e2-1.1-c19e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $2308.94$
Root an. cond. $6.93191$
Motivic weight $19$
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  + 5.90e4·3-s − 5.24e5·4-s + 9.68e7·7-s + 2.32e9·9-s − 3.09e10·12-s + 2.04e12·19-s + 5.71e12·21-s + 1.90e13·25-s + 6.86e13·27-s − 5.07e13·28-s − 4.79e14·31-s − 1.21e15·36-s + 1.58e15·37-s + 1.20e16·43-s − 2.01e15·49-s + 1.20e17·57-s + 2.02e17·61-s + 2.25e17·63-s + 1.44e17·64-s + 4.41e17·67-s − 2.86e17·73-s + 1.12e18·75-s − 1.07e18·76-s − 2.62e17·79-s + 1.35e18·81-s − 2.99e18·84-s − 2.82e19·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 0.907·7-s + 2·9-s − 1.73·12-s + 1.45·19-s + 1.57·21-s + 25-s + 1.73·27-s − 0.907·28-s − 3.25·31-s − 2·36-s + 1.99·37-s + 3.64·43-s − 0.177·49-s + 2.52·57-s + 2.21·61-s + 1.81·63-s + 64-s + 1.98·67-s − 0.569·73-s + 1.73·75-s − 1.45·76-s − 0.246·79-s + 81-s − 1.57·84-s − 5.63·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2308.94\)
Root analytic conductor: \(6.93191\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(6.277179504\)
\(L(\frac12)\) \(\approx\) \(6.277179504\)
\(L(\frac{21}{2})\) 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$C_2$ \( 1 - p^{10} T + p^{19} T^{2} \)
7$C_2$ \( 1 - 96856453 T + p^{19} T^{2} \)
good2$C_2^2$ \( 1 + p^{19} T^{2} + p^{38} T^{4} \)
5$C_2^2$ \( 1 - p^{19} T^{2} + p^{38} T^{4} \)
11$C_2^2$ \( 1 + p^{19} T^{2} + p^{38} T^{4} \)
13$C_2$ \( ( 1 - 7913874521 T + p^{19} T^{2} )( 1 + 7913874521 T + p^{19} T^{2} ) \)
17$C_2^2$ \( 1 - p^{19} T^{2} + p^{38} T^{4} \)
19$C_2$ \( ( 1 - 2300181624971 T + p^{19} T^{2} )( 1 + 252454137272 T + p^{19} T^{2} ) \)
23$C_2^2$ \( 1 + p^{19} T^{2} + p^{38} T^{4} \)
29$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 189181140858356 T + p^{19} T^{2} )( 1 + 289961649705409 T + p^{19} T^{2} ) \)
37$C_2$ \( ( 1 - 800421296193277 T + p^{19} T^{2} )( 1 - 780590048220370 T + p^{19} T^{2} ) \)
41$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6012529514786335 T + p^{19} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{19} T^{2} + p^{38} T^{4} \)
53$C_2^2$ \( 1 + p^{19} T^{2} + p^{38} T^{4} \)
59$C_2^2$ \( 1 - p^{19} T^{2} + p^{38} T^{4} \)
61$C_2$ \( ( 1 - 171366248956165199 T + p^{19} T^{2} )( 1 - 30901601965099333 T + p^{19} T^{2} ) \)
67$C_2$ \( ( 1 - 273531389481429392 T + p^{19} T^{2} )( 1 - 167643129887418785 T + p^{19} T^{2} ) \)
71$C_2$ \( ( 1 - p^{19} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 352850417428850191 T + p^{19} T^{2} )( 1 + 639446467598381530 T + p^{19} T^{2} ) \)
79$C_2$ \( ( 1 - 1699851029529528053 T + p^{19} T^{2} )( 1 + 1962240154233379276 T + p^{19} T^{2} ) \)
83$C_2$ \( ( 1 + p^{19} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{19} T^{2} + p^{38} T^{4} \)
97$C_2$ \( ( 1 - 9502640950925166898 T + p^{19} T^{2} )( 1 + 9502640950925166898 T + p^{19} T^{2} ) \)
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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

−14.21588995834844244517892802251, −13.64612183099239223210685742333, −12.78414203779864028188276129199, −12.73358025340843565084821453713, −11.31751567418832083638585903401, −10.89251528305971152572764262783, −9.642439526416713733468941961207, −9.400428685343997025739366777072, −8.861300666120112080790741056391, −8.201421185858428547167459474895, −7.54060035481372786026296431734, −7.13682848223445912008143925225, −5.63023582551122440147807062809, −4.99621583612341112379524321898, −4.07941032490452674832954258172, −3.78894834793450174241616401295, −2.75231405181806532155629499226, −2.19353664885276320573602748104, −1.26911196060362356602565407853, −0.67109127058146114767755007230, 0.67109127058146114767755007230, 1.26911196060362356602565407853, 2.19353664885276320573602748104, 2.75231405181806532155629499226, 3.78894834793450174241616401295, 4.07941032490452674832954258172, 4.99621583612341112379524321898, 5.63023582551122440147807062809, 7.13682848223445912008143925225, 7.54060035481372786026296431734, 8.201421185858428547167459474895, 8.861300666120112080790741056391, 9.400428685343997025739366777072, 9.642439526416713733468941961207, 10.89251528305971152572764262783, 11.31751567418832083638585903401, 12.73358025340843565084821453713, 12.78414203779864028188276129199, 13.64612183099239223210685742333, 14.21588995834844244517892802251

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