Properties

Label 4-945e2-1.1-c1e2-0-66
Degree $4$
Conductor $893025$
Sign $1$
Analytic cond. $56.9400$
Root an. cond. $2.74697$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·4-s + 2·5-s − 2·7-s + 3·8-s − 2·10-s − 2·11-s − 13-s + 2·14-s + 16-s − 11·17-s + 3·19-s − 4·20-s + 2·22-s − 11·23-s + 3·25-s + 26-s + 4·28-s + 29-s + 6·31-s − 2·32-s + 11·34-s − 4·35-s − 6·37-s − 3·38-s + 6·40-s − 41-s + ⋯
L(s)  = 1  − 0.707·2-s − 4-s + 0.894·5-s − 0.755·7-s + 1.06·8-s − 0.632·10-s − 0.603·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 2.66·17-s + 0.688·19-s − 0.894·20-s + 0.426·22-s − 2.29·23-s + 3/5·25-s + 0.196·26-s + 0.755·28-s + 0.185·29-s + 1.07·31-s − 0.353·32-s + 1.88·34-s − 0.676·35-s − 0.986·37-s − 0.486·38-s + 0.948·40-s − 0.156·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(893025\)    =    \(3^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(56.9400\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 893025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 11 T + 75 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - T + 47 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 20 T + 181 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 65 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 11 T + 135 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 89 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 11 T + 161 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 14 T + 175 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 239 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 3 T - 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.768959638262355209845269176243, −9.550660287951508541321813423167, −9.044804327219182083733636154064, −8.566531395598523348539171568346, −8.369830254285220252696450966937, −8.041983683939172592327563607475, −7.29344693568624867780204956350, −6.82249740246230560233059689989, −6.34580795642254139215574070577, −6.19626267906589685771942440380, −5.37463798067821806922003042876, −5.02573903522813994495500054452, −4.41379801848985815164581208006, −4.26981627676755850987712579643, −3.31625884416302874363695905680, −2.85727326491296238649623992137, −2.03468494786555335800175458069, −1.61246665603708375641685751921, 0, 0, 1.61246665603708375641685751921, 2.03468494786555335800175458069, 2.85727326491296238649623992137, 3.31625884416302874363695905680, 4.26981627676755850987712579643, 4.41379801848985815164581208006, 5.02573903522813994495500054452, 5.37463798067821806922003042876, 6.19626267906589685771942440380, 6.34580795642254139215574070577, 6.82249740246230560233059689989, 7.29344693568624867780204956350, 8.041983683939172592327563607475, 8.369830254285220252696450966937, 8.566531395598523348539171568346, 9.044804327219182083733636154064, 9.550660287951508541321813423167, 9.768959638262355209845269176243

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