Properties

Label 4-45e2-1.1-c7e2-0-3
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $197.608$
Root an. cond. $3.74931$
Motivic weight $7$
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  − 7·2-s − 69·4-s − 250·5-s + 1.30e3·7-s + 413·8-s + 1.75e3·10-s − 3.44e3·11-s − 8.98e3·13-s − 9.12e3·14-s − 4.86e3·16-s + 5.49e3·17-s − 4.95e4·19-s + 1.72e4·20-s + 2.41e4·22-s − 9.18e4·23-s + 4.68e4·25-s + 6.29e4·26-s − 8.99e4·28-s − 1.81e5·29-s + 3.04e5·31-s + 1.61e5·32-s − 3.84e4·34-s − 3.26e5·35-s − 5.02e5·37-s + 3.47e5·38-s − 1.03e5·40-s − 6.31e5·41-s + ⋯
L(s)  = 1  − 0.618·2-s − 0.539·4-s − 0.894·5-s + 1.43·7-s + 0.285·8-s + 0.553·10-s − 0.781·11-s − 1.13·13-s − 0.889·14-s − 0.296·16-s + 0.271·17-s − 1.65·19-s + 0.482·20-s + 0.483·22-s − 1.57·23-s + 3/5·25-s + 0.702·26-s − 0.774·28-s − 1.38·29-s + 1.83·31-s + 0.872·32-s − 0.167·34-s − 1.28·35-s − 1.63·37-s + 1.02·38-s − 0.255·40-s − 1.43·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(197.608\)
Root analytic conductor: \(3.74931\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2025,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 + p^{3} T )^{2} \)
good2$D_{4}$ \( 1 + 7 T + 59 p T^{2} + 7 p^{7} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 - 1304 T + 1601006 T^{2} - 1304 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 + 3448 T + 9598294 T^{2} + 3448 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 8988 T + 43676926 T^{2} + 8988 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 5492 T + 533727862 T^{2} - 5492 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 + 49584 T + 1767259558 T^{2} + 49584 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 + 91848 T + 4843394254 T^{2} + 91848 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 6268 p T + 32439203278 T^{2} + 6268 p^{8} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 304232 T + 77458297022 T^{2} - 304232 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 502316 T + 221684315886 T^{2} + 502316 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 + 631172 T + 420346017142 T^{2} + 631172 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 353640 T + 567251429590 T^{2} - 353640 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 467480 T + 1062629128990 T^{2} - 467480 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 - 568052 T + 2403786403294 T^{2} - 568052 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 + 287224 T + 1627391637238 T^{2} + 287224 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 2514180 T + 7865442419758 T^{2} + 2514180 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 + 5073832 T + 16021265240102 T^{2} + 5073832 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 3748816 T + 20824804809646 T^{2} - 3748816 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 1477212 T - 3158190986 T^{2} + 1477212 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 + 4627720 T + 42789559383518 T^{2} + 4627720 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 6072936 T + 62224060120582 T^{2} - 6072936 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 + 16516356 T + 156597055746838 T^{2} + 16516356 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 2723428 T + 135845063471622 T^{2} - 2723428 p^{7} T^{3} + p^{14} T^{4} \)
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

−13.84188907858769980231030012621, −13.81849805991726470654828166429, −12.64154293168540013826529166892, −12.22829697892357241591768583173, −11.64710674077183346086845416618, −11.03745075735779380247780328528, −10.27130159313228053604390998013, −9.939056423156428691375938383141, −8.734492645305948054844981074791, −8.554147740585570774472803405260, −7.81092460717480919964266114578, −7.47085272984639799998415172378, −6.34083011669719464819863907502, −5.20345335328960321441633260041, −4.61207387956342506577495486947, −4.00798575400841684145448791088, −2.57140154513953339983378026514, −1.62901188374473814017497986832, 0, 0, 1.62901188374473814017497986832, 2.57140154513953339983378026514, 4.00798575400841684145448791088, 4.61207387956342506577495486947, 5.20345335328960321441633260041, 6.34083011669719464819863907502, 7.47085272984639799998415172378, 7.81092460717480919964266114578, 8.554147740585570774472803405260, 8.734492645305948054844981074791, 9.939056423156428691375938383141, 10.27130159313228053604390998013, 11.03745075735779380247780328528, 11.64710674077183346086845416618, 12.22829697892357241591768583173, 12.64154293168540013826529166892, 13.81849805991726470654828166429, 13.84188907858769980231030012621

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