Properties

Label 4-21e2-1.1-c35e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $26552.6$
Root an. cond. $12.7651$
Motivic weight $35$
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  − 3.87e8·3-s − 3.43e10·4-s − 4.46e14·7-s + 1.00e17·9-s + 1.33e19·12-s − 2.90e22·19-s + 1.72e23·21-s + 2.91e24·25-s − 1.93e25·27-s + 1.53e25·28-s + 9.92e25·31-s − 3.43e27·36-s + 1.43e27·37-s + 1.38e29·43-s − 1.79e29·49-s + 1.12e31·57-s + 2.22e31·61-s − 4.46e31·63-s + 4.05e31·64-s + 1.10e32·67-s + 8.33e32·73-s − 1.12e33·75-s + 9.97e32·76-s + 1.35e33·79-s + 2.50e33·81-s − 5.94e33·84-s − 3.84e34·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 0.725·7-s + 2·9-s + 1.73·12-s − 1.21·19-s + 1.25·21-s + 25-s − 1.73·27-s + 0.725·28-s + 0.790·31-s − 2·36-s + 0.516·37-s + 3.60·43-s − 0.473·49-s + 2.10·57-s + 1.27·61-s − 1.45·63-s + 64-s + 1.21·67-s + 2.05·73-s − 1.73·75-s + 1.21·76-s + 0.840·79-s + 81-s − 1.25·84-s − 1.36·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+35/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26552.6\)
Root analytic conductor: \(12.7651\)
Motivic weight: \(35\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :35/2, 35/2),\ 1)\)

Particular Values

\(L(18)\) \(\approx\) \(0.5293963146\)
\(L(\frac12)\) \(\approx\) \(0.5293963146\)
\(L(\frac{37}{2})\) 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$C_2$ \( 1 + p^{18} T + p^{35} T^{2} \)
7$C_2$ \( 1 + 446525205377873 T + p^{35} T^{2} \)
good2$C_2^2$ \( 1 + p^{35} T^{2} + p^{70} T^{4} \)
5$C_2^2$ \( 1 - p^{35} T^{2} + p^{70} T^{4} \)
11$C_2^2$ \( 1 + p^{35} T^{2} + p^{70} T^{4} \)
13$C_2$ \( ( 1 - 37039694716780372589 T + p^{35} T^{2} )( 1 + 37039694716780372589 T + p^{35} T^{2} ) \)
17$C_2^2$ \( 1 - p^{35} T^{2} + p^{70} T^{4} \)
19$C_2$ \( ( 1 - \)\(78\!\cdots\!68\)\( T + p^{35} T^{2} )( 1 + \)\(36\!\cdots\!39\)\( T + p^{35} T^{2} ) \)
23$C_2^2$ \( 1 + p^{35} T^{2} + p^{70} T^{4} \)
29$C_2$ \( ( 1 - p^{35} T^{2} )^{2} \)
31$C_2$ \( ( 1 - \)\(17\!\cdots\!01\)\( T + p^{35} T^{2} )( 1 + \)\(72\!\cdots\!76\)\( T + p^{35} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(53\!\cdots\!73\)\( T + p^{35} T^{2} )( 1 + \)\(39\!\cdots\!50\)\( T + p^{35} T^{2} ) \)
41$C_2$ \( ( 1 + p^{35} T^{2} )^{2} \)
43$C_2$ \( ( 1 - \)\(69\!\cdots\!25\)\( T + p^{35} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{35} T^{2} + p^{70} T^{4} \)
53$C_2^2$ \( 1 + p^{35} T^{2} + p^{70} T^{4} \)
59$C_2^2$ \( 1 - p^{35} T^{2} + p^{70} T^{4} \)
61$C_2$ \( ( 1 - \)\(27\!\cdots\!99\)\( T + p^{35} T^{2} )( 1 + \)\(51\!\cdots\!27\)\( T + p^{35} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(17\!\cdots\!75\)\( T + p^{35} T^{2} )( 1 + \)\(69\!\cdots\!52\)\( T + p^{35} T^{2} ) \)
71$C_2$ \( ( 1 - p^{35} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(74\!\cdots\!39\)\( T + p^{35} T^{2} )( 1 - \)\(90\!\cdots\!50\)\( T + p^{35} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(32\!\cdots\!64\)\( T + p^{35} T^{2} )( 1 + \)\(18\!\cdots\!57\)\( T + p^{35} T^{2} ) \)
83$C_2$ \( ( 1 + p^{35} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{35} T^{2} + p^{70} T^{4} \)
97$C_2$ \( ( 1 - \)\(66\!\cdots\!02\)\( T + p^{35} T^{2} )( 1 + \)\(66\!\cdots\!02\)\( T + p^{35} T^{2} ) \)
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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

−11.73483920741757462634124708152, −10.95323708837163594021090103007, −10.77821656325845291539486385410, −10.01359475385188510441448479298, −9.497922675750068443263982667116, −8.970623294105340148626751275444, −8.260511658592898329462771415867, −7.45741552922853064823835045939, −6.74807217142832369620005303356, −6.30330192552762989897248401937, −5.91231097111558047840653327116, −4.96841081845382692804766035282, −4.92184650648417052024467218192, −3.97791156693372102060613351295, −3.89181966573401014847758825833, −2.65047310455463023216562536954, −2.18529138589837449068738895129, −0.942008023110867236384229169918, −0.899881442385769637931273516010, −0.24345685328135260614960632439, 0.24345685328135260614960632439, 0.899881442385769637931273516010, 0.942008023110867236384229169918, 2.18529138589837449068738895129, 2.65047310455463023216562536954, 3.89181966573401014847758825833, 3.97791156693372102060613351295, 4.92184650648417052024467218192, 4.96841081845382692804766035282, 5.91231097111558047840653327116, 6.30330192552762989897248401937, 6.74807217142832369620005303356, 7.45741552922853064823835045939, 8.260511658592898329462771415867, 8.970623294105340148626751275444, 9.497922675750068443263982667116, 10.01359475385188510441448479298, 10.77821656325845291539486385410, 10.95323708837163594021090103007, 11.73483920741757462634124708152

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