Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s + 4·11-s + 6·12-s + 6·13-s + 5·16-s + 6·17-s − 6·18-s − 8·22-s + 2·23-s − 8·24-s − 12·26-s + 4·27-s + 2·29-s − 2·31-s − 6·32-s + 8·33-s − 12·34-s + 9·36-s + 12·39-s − 6·41-s + 6·43-s + 12·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s + 1.20·11-s + 1.73·12-s + 1.66·13-s + 5/4·16-s + 1.45·17-s − 1.41·18-s − 1.70·22-s + 0.417·23-s − 1.63·24-s − 2.35·26-s + 0.769·27-s + 0.371·29-s − 0.359·31-s − 1.06·32-s + 1.39·33-s − 2.05·34-s + 3/2·36-s + 1.92·39-s − 0.937·41-s + 0.914·43-s + 1.80·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(54022500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7350} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 54022500,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.576085698\)
\(L(\frac12)\)  \(\approx\)  \(4.576085698\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good11$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 2 T + 61 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 93 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 123 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 160 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 144 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.041555851186913625288322312294, −7.897701697077876066231832078797, −7.44609286087223495405952354758, −7.31193330571970690133007942501, −6.69733594249245796766209872567, −6.61747002678519463358507992263, −6.00354302515484419868841885436, −5.99100082271201054162744445338, −5.19454460082498916006587547631, −5.14136987488770864214114403276, −4.13248206154617541701432238752, −4.02474321001335692229011899624, −3.51982733141073725737441165224, −3.45478792375148122724305579183, −2.65931115796620108641636662602, −2.59448175784176780295541620585, −1.73016443050541720897329165760, −1.57346745670362878041009768221, −0.922714190537144890624526445936, −0.78379502172290067623818240660, 0.78379502172290067623818240660, 0.922714190537144890624526445936, 1.57346745670362878041009768221, 1.73016443050541720897329165760, 2.59448175784176780295541620585, 2.65931115796620108641636662602, 3.45478792375148122724305579183, 3.51982733141073725737441165224, 4.02474321001335692229011899624, 4.13248206154617541701432238752, 5.14136987488770864214114403276, 5.19454460082498916006587547631, 5.99100082271201054162744445338, 6.00354302515484419868841885436, 6.61747002678519463358507992263, 6.69733594249245796766209872567, 7.31193330571970690133007942501, 7.44609286087223495405952354758, 7.897701697077876066231832078797, 8.041555851186913625288322312294

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