Properties

Label 4-230e2-1.1-c3e2-0-0
Degree $4$
Conductor $52900$
Sign $1$
Analytic cond. $184.156$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 3·3-s + 12·4-s + 10·5-s + 12·6-s − 17·7-s − 32·8-s − 29·9-s − 40·10-s + 18·11-s − 36·12-s − 9·13-s + 68·14-s − 30·15-s + 80·16-s − 79·17-s + 116·18-s + 34·19-s + 120·20-s + 51·21-s − 72·22-s − 46·23-s + 96·24-s + 75·25-s + 36·26-s + 120·27-s − 204·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 0.917·7-s − 1.41·8-s − 1.07·9-s − 1.26·10-s + 0.493·11-s − 0.866·12-s − 0.192·13-s + 1.29·14-s − 0.516·15-s + 5/4·16-s − 1.12·17-s + 1.51·18-s + 0.410·19-s + 1.34·20-s + 0.529·21-s − 0.697·22-s − 0.417·23-s + 0.816·24-s + 3/5·25-s + 0.271·26-s + 0.855·27-s − 1.37·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(52900\)    =    \(2^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(184.156\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{230} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 52900,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
5$C_1$ \( ( 1 - p T )^{2} \)
23$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 + p T + 38 T^{2} + p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 17 T + 740 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 18 T + 918 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 9 T + 2936 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 79 T + 9178 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 34 T + 12182 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 112 T + 51841 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 92 T + 49361 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 491 T + 160682 T^{2} + 491 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 332 T + 65461 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 354 T + 181510 T^{2} + 354 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 599 T + 296890 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 305 T + 320116 T^{2} + 305 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 357 T + 241414 T^{2} - 357 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 172 T + 458730 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 531 T + 641338 T^{2} + 531 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1254 T + 911559 T^{2} - 1254 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 343 T + 797792 T^{2} + 343 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 88 T + 930782 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1273 T + 1298882 T^{2} - 1273 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1106 T + 1709834 T^{2} - 1106 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 2240 T + 2588894 T^{2} + 2240 p^{3} T^{3} + p^{6} T^{4} \)
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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

−11.32277614476745674035118807367, −11.06261969262693208466955536733, −10.33377867759958170136745116282, −10.10890232067053498085827224232, −9.371211236571410539328900317643, −9.341279733082870591753859791922, −8.581279755219757847558971592284, −8.350720908631529229360829593159, −7.53128142988758434896139068843, −6.77430176058853290361448310050, −6.44591785665238699526585567015, −6.26064044448570022947235220788, −5.26476880772818464526443845018, −5.08285235786245020151215987983, −3.54923107364091131856081569418, −3.14432059586994515024987533966, −2.11678931097840996637049498027, −1.56218787078874294915871330238, 0, 0, 1.56218787078874294915871330238, 2.11678931097840996637049498027, 3.14432059586994515024987533966, 3.54923107364091131856081569418, 5.08285235786245020151215987983, 5.26476880772818464526443845018, 6.26064044448570022947235220788, 6.44591785665238699526585567015, 6.77430176058853290361448310050, 7.53128142988758434896139068843, 8.350720908631529229360829593159, 8.581279755219757847558971592284, 9.341279733082870591753859791922, 9.371211236571410539328900317643, 10.10890232067053498085827224232, 10.33377867759958170136745116282, 11.06261969262693208466955536733, 11.32277614476745674035118807367

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