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Properties

Label 4-18e2-1.1-c25e2-0-2
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $5080.75$
Root an. cond. $8.44271$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.19e3·2-s + 5.03e7·4-s − 6.97e8·5-s − 1.92e10·7-s − 2.74e11·8-s + 5.71e12·10-s + 5.75e12·11-s + 7.01e13·13-s + 1.57e14·14-s + 1.40e15·16-s − 3.13e15·17-s + 1.60e16·19-s − 3.51e16·20-s − 4.71e16·22-s − 8.23e16·23-s − 1.73e17·25-s − 5.74e17·26-s − 9.70e17·28-s + 2.01e18·29-s + 7.61e18·31-s − 6.91e18·32-s + 2.57e19·34-s + 1.34e19·35-s + 3.70e19·37-s − 1.31e20·38-s + 1.91e20·40-s − 1.46e20·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.27·5-s − 0.526·7-s − 1.41·8-s + 1.80·10-s + 0.552·11-s + 0.834·13-s + 0.744·14-s + 5/4·16-s − 1.30·17-s + 1.66·19-s − 1.91·20-s − 0.781·22-s − 0.783·23-s − 0.583·25-s − 1.18·26-s − 0.789·28-s + 1.05·29-s + 1.73·31-s − 1.06·32-s + 1.84·34-s + 0.673·35-s + 0.924·37-s − 2.35·38-s + 1.80·40-s − 1.01·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(5080.75\)
Root analytic conductor: \(8.44271\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 324,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) 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)$
bad2$C_1$ \( ( 1 + p^{12} T )^{2} \)
3 \( 1 \)
good5$D_{4}$ \( 1 + 697960704 T + 26445212384474074 p^{2} T^{2} + 697960704 p^{25} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 + 2755057400 p T - 29167885836553890 p^{4} T^{2} + 2755057400 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 - 5751609553920 T + \)\(31\!\cdots\!46\)\( p^{2} T^{2} - 5751609553920 p^{25} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 - 70123116734020 T + \)\(11\!\cdots\!50\)\( p T^{2} - 70123116734020 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 + 3138018908355072 T + \)\(44\!\cdots\!30\)\( p T^{2} + 3138018908355072 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 - 843826548954928 p T + \)\(42\!\cdots\!14\)\( p^{2} T^{2} - 843826548954928 p^{26} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 + 3580322113812480 p T + \)\(28\!\cdots\!34\)\( p^{2} T^{2} + 3580322113812480 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 - 2013962822921222400 T + \)\(72\!\cdots\!82\)\( T^{2} - 2013962822921222400 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 - 7619316290120600632 T + \)\(51\!\cdots\!58\)\( T^{2} - 7619316290120600632 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 - 37028608192780653100 T + \)\(22\!\cdots\!10\)\( T^{2} - 37028608192780653100 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 + \)\(14\!\cdots\!40\)\( T + \)\(39\!\cdots\!02\)\( T^{2} + \)\(14\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 - \)\(53\!\cdots\!60\)\( T + \)\(20\!\cdots\!82\)\( T^{2} - \)\(53\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(11\!\cdots\!14\)\( T^{2} - \)\(15\!\cdots\!00\)\( p^{25} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 + \)\(29\!\cdots\!12\)\( T + \)\(20\!\cdots\!22\)\( T^{2} + \)\(29\!\cdots\!12\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 + \)\(16\!\cdots\!40\)\( T + \)\(37\!\cdots\!22\)\( T^{2} + \)\(16\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 + \)\(21\!\cdots\!56\)\( T + \)\(97\!\cdots\!86\)\( T^{2} + \)\(21\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 + \)\(49\!\cdots\!00\)\( T + \)\(86\!\cdots\!78\)\( T^{2} + \)\(49\!\cdots\!00\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(31\!\cdots\!78\)\( T^{2} + \)\(11\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 + \)\(12\!\cdots\!00\)\( p T + \)\(77\!\cdots\!86\)\( T^{2} + \)\(12\!\cdots\!00\)\( p^{26} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 - \)\(72\!\cdots\!32\)\( T + \)\(39\!\cdots\!54\)\( T^{2} - \)\(72\!\cdots\!32\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 + \)\(35\!\cdots\!36\)\( T + \)\(51\!\cdots\!10\)\( T^{2} + \)\(35\!\cdots\!36\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 + \)\(43\!\cdots\!80\)\( T + \)\(12\!\cdots\!74\)\( T^{2} + \)\(43\!\cdots\!80\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 + \)\(64\!\cdots\!60\)\( T + \)\(50\!\cdots\!50\)\( T^{2} + \)\(64\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} \)
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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

−12.17574027772922859028775042965, −12.16857191608576675747375282873, −11.23942035160694802013347858713, −11.05914631201122350530554091002, −9.947591976128072252132638912847, −9.621945018711544479692308274349, −8.772678765511116516618880066758, −8.293575703547210447642695186600, −7.66714899887455119881170616081, −7.12524480227301529836037151204, −6.31833240093771047831713269010, −5.87837613027414417197182175531, −4.43204345470118202916086025880, −3.99693323622267867168920605900, −3.05781314965292429023807515719, −2.60724810410190481077920319615, −1.38208531782348961054387952532, −1.08290307291920181074432832921, 0, 0, 1.08290307291920181074432832921, 1.38208531782348961054387952532, 2.60724810410190481077920319615, 3.05781314965292429023807515719, 3.99693323622267867168920605900, 4.43204345470118202916086025880, 5.87837613027414417197182175531, 6.31833240093771047831713269010, 7.12524480227301529836037151204, 7.66714899887455119881170616081, 8.293575703547210447642695186600, 8.772678765511116516618880066758, 9.621945018711544479692308274349, 9.947591976128072252132638912847, 11.05914631201122350530554091002, 11.23942035160694802013347858713, 12.16857191608576675747375282873, 12.17574027772922859028775042965

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