Properties

Label 4-210e2-1.1-c7e2-0-4
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $4303.47$
Root an. cond. $8.09943$
Motivic weight $7$
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  + 16·2-s − 54·3-s + 192·4-s − 250·5-s − 864·6-s + 686·7-s + 2.04e3·8-s + 2.18e3·9-s − 4.00e3·10-s + 4.09e3·11-s − 1.03e4·12-s − 7.28e3·13-s + 1.09e4·14-s + 1.35e4·15-s + 2.04e4·16-s − 1.38e4·17-s + 3.49e4·18-s − 4.00e3·19-s − 4.80e4·20-s − 3.70e4·21-s + 6.54e4·22-s − 9.09e3·23-s − 1.10e5·24-s + 4.68e4·25-s − 1.16e5·26-s − 7.87e4·27-s + 1.31e5·28-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 + 0.926·11-s − 1.73·12-s − 0.919·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.684·17-s + 1.41·18-s − 0.133·19-s − 1.34·20-s − 0.872·21-s + 1.31·22-s − 0.155·23-s − 1.63·24-s + 3/5·25-s − 1.29·26-s − 0.769·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4303.47\)
Root analytic conductor: \(8.09943\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(6.741920048\)
\(L(\frac12)\) \(\approx\) \(6.741920048\)
\(L(\frac{9}{2})\) 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)$
bad2$C_1$ \( ( 1 - p^{3} T )^{2} \)
3$C_1$ \( ( 1 + p^{3} T )^{2} \)
5$C_1$ \( ( 1 + p^{3} T )^{2} \)
7$C_1$ \( ( 1 - p^{3} T )^{2} \)
good11$D_{4}$ \( 1 - 372 p T + 40987894 T^{2} - 372 p^{8} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 + 560 p T + 32290998 T^{2} + 560 p^{8} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 + 13860 T + 729658150 T^{2} + 13860 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 + 4004 T - 574170714 T^{2} + 4004 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 + 9096 T + 6274158814 T^{2} + 9096 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 34740 T + 17977133902 T^{2} + 34740 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 + 29852 T + 49597174734 T^{2} + 29852 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 - 442708 T + 233429937582 T^{2} - 442708 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 162108 T + 205726931254 T^{2} - 162108 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 1091872 T + 836251928310 T^{2} - 1091872 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 1630968 T + 29515583234 p T^{2} - 1630968 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 - 2146848 T + 2982155687206 T^{2} - 2146848 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 2106624 T + 5808324646486 T^{2} - 2106624 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 1528532 T + 5640994558542 T^{2} + 1528532 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 1987240 T + 7767185923110 T^{2} - 1987240 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 5059644 T + 24514267277950 T^{2} - 5059644 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 1321568 T + 10048729261374 T^{2} + 1321568 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 11493496 T + 67138127820318 T^{2} - 11493496 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 4920816 T + 52466233289062 T^{2} - 4920816 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 12159228 T + 125190643054678 T^{2} - 12159228 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 4037080 T + 85227056632926 T^{2} - 4037080 p^{7} T^{3} + p^{14} T^{4} \)
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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

−11.43718806273661410070332306890, −11.17225229144460574444568182153, −10.61057828648467254119015545277, −10.32571695469487607255232180417, −9.285876860803882559282479935865, −9.042715626834776641785974720120, −7.991679933937344328609936645240, −7.60601981985768951342004776223, −7.14648731586040380335303970952, −6.65656241484393953180725395718, −6.03269048374448291497771267452, −5.59538284014113899204212652153, −4.92198761566172100386480607529, −4.54033588240734482783719700716, −3.96107765067658904154487449049, −3.73158455790817752638941197940, −2.33278430587191177722371264034, −2.24466024350603765609431978603, −0.827614539916170250218743815844, −0.76474224831916151252463826841, 0.76474224831916151252463826841, 0.827614539916170250218743815844, 2.24466024350603765609431978603, 2.33278430587191177722371264034, 3.73158455790817752638941197940, 3.96107765067658904154487449049, 4.54033588240734482783719700716, 4.92198761566172100386480607529, 5.59538284014113899204212652153, 6.03269048374448291497771267452, 6.65656241484393953180725395718, 7.14648731586040380335303970952, 7.60601981985768951342004776223, 7.991679933937344328609936645240, 9.042715626834776641785974720120, 9.285876860803882559282479935865, 10.32571695469487607255232180417, 10.61057828648467254119015545277, 11.17225229144460574444568182153, 11.43718806273661410070332306890

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