Properties

Label 4-210e2-1.1-c5e2-0-3
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $1134.38$
Root an. cond. $5.80349$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 18·3-s + 48·4-s − 50·5-s + 144·6-s + 98·7-s + 256·8-s + 243·9-s − 400·10-s + 102·11-s + 864·12-s + 574·13-s + 784·14-s − 900·15-s + 1.28e3·16-s + 1.36e3·17-s + 1.94e3·18-s + 1.84e3·19-s − 2.40e3·20-s + 1.76e3·21-s + 816·22-s + 816·23-s + 4.60e3·24-s + 1.87e3·25-s + 4.59e3·26-s + 2.91e3·27-s + 4.70e3·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s + 0.254·11-s + 1.73·12-s + 0.942·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.14·17-s + 1.41·18-s + 1.17·19-s − 1.34·20-s + 0.872·21-s + 0.359·22-s + 0.321·23-s + 1.63·24-s + 3/5·25-s + 1.33·26-s + 0.769·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1134.38\)
Root analytic conductor: \(5.80349\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(16.59986042\)
\(L(\frac12)\) \(\approx\) \(16.59986042\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
3$C_1$ \( ( 1 - p^{2} T )^{2} \)
5$C_1$ \( ( 1 + p^{2} T )^{2} \)
7$C_1$ \( ( 1 - p^{2} T )^{2} \)
good11$D_{4}$ \( 1 - 102 T + 80182 T^{2} - 102 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 574 T + 580434 T^{2} - 574 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 1368 T + 2329486 T^{2} - 1368 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 1846 T + 3603438 T^{2} - 1846 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 816 T - 2610194 T^{2} - 816 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 4824 T + 45861958 T^{2} - 4824 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 7738 T + 70026774 T^{2} - 7738 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 14740 T + 177355470 T^{2} - 14740 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 2808 T + 115335454 T^{2} - 2808 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 11668 T + 209704278 T^{2} - 11668 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 14676 T + 292467358 T^{2} + 14676 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 15630 T + 525548770 T^{2} + 15630 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 23412 T + 1049472598 T^{2} + 23412 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 30416 T + 1309173366 T^{2} + 30416 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 9836 T + 2699984838 T^{2} + 9836 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 49422 T + 3367914622 T^{2} - 49422 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 19718 T + 2867912442 T^{2} + 19718 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 92344 T + 7624781598 T^{2} - 92344 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 42864 T + 6612071734 T^{2} - 42864 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 30516 T + 2758575238 T^{2} - 30516 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 123122 T + 14467758786 T^{2} + 123122 p^{5} T^{3} + p^{10} T^{4} \)
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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

−11.65821712648024614233019675636, −11.65085353836648406320170762392, −10.75919980620413626794264224024, −10.64158675861256097433311697104, −9.627990193473713379642969962971, −9.402979117626694866628831507827, −8.482357927169687884164137006947, −7.995520025235975874919433084066, −7.81548865924773002604568104691, −7.31041775944726369953240650959, −6.42717654601313499113636841696, −6.16001977872682795153205618051, −5.06708884727054414311810891508, −4.85281540320257861727362361683, −3.95860821630095553108061687848, −3.78519700915971230496443589493, −2.82718446291030073290753477871, −2.73678657647932157641128242629, −1.21599110677607789911023799172, −1.18408646729031093850532956026, 1.18408646729031093850532956026, 1.21599110677607789911023799172, 2.73678657647932157641128242629, 2.82718446291030073290753477871, 3.78519700915971230496443589493, 3.95860821630095553108061687848, 4.85281540320257861727362361683, 5.06708884727054414311810891508, 6.16001977872682795153205618051, 6.42717654601313499113636841696, 7.31041775944726369953240650959, 7.81548865924773002604568104691, 7.995520025235975874919433084066, 8.482357927169687884164137006947, 9.402979117626694866628831507827, 9.627990193473713379642969962971, 10.64158675861256097433311697104, 10.75919980620413626794264224024, 11.65085353836648406320170762392, 11.65821712648024614233019675636

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