Properties

Label 4-5290e2-1.1-c1e2-0-10
Degree $4$
Conductor $27984100$
Sign $1$
Analytic cond. $1784.29$
Root an. cond. $6.49929$
Motivic weight $1$
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  + 2·2-s + 3-s + 3·4-s − 2·5-s + 2·6-s − 7-s + 4·8-s − 4·9-s − 4·10-s − 11-s + 3·12-s − 3·13-s − 2·14-s − 2·15-s + 5·16-s − 17-s − 8·18-s + 3·19-s − 6·20-s − 21-s − 2·22-s + 4·24-s + 3·25-s − 6·26-s − 6·27-s − 3·28-s − 14·29-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s + 0.816·6-s − 0.377·7-s + 1.41·8-s − 4/3·9-s − 1.26·10-s − 0.301·11-s + 0.866·12-s − 0.832·13-s − 0.534·14-s − 0.516·15-s + 5/4·16-s − 0.242·17-s − 1.88·18-s + 0.688·19-s − 1.34·20-s − 0.218·21-s − 0.426·22-s + 0.816·24-s + 3/5·25-s − 1.17·26-s − 1.15·27-s − 0.566·28-s − 2.59·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27984100\)    =    \(2^{2} \cdot 5^{2} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1784.29\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27984100,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
23 \( 1 \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 63 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 27 T + 375 T^{2} + 27 p T^{3} + p^{2} T^{4} \)
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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

−7.81906273187618179676538576740, −7.75274728439861315120651588333, −7.08648667788623873489791526150, −7.08270744356990329242129282345, −6.56558165991719062802854989718, −6.15620996062020129395201056662, −5.54381359289116914041628820766, −5.52344065907571606240257717444, −5.19703992661925957481520918925, −4.64950704676480919580323590583, −4.17903409931898013180558604652, −3.99633771144638777938101362794, −3.27545247173021532503507514505, −3.25218831129758247552551985199, −2.80966264078545540705206855109, −2.47777112853799117885369613054, −1.85854846336723654086047647703, −1.33396645744727983997076155033, 0, 0, 1.33396645744727983997076155033, 1.85854846336723654086047647703, 2.47777112853799117885369613054, 2.80966264078545540705206855109, 3.25218831129758247552551985199, 3.27545247173021532503507514505, 3.99633771144638777938101362794, 4.17903409931898013180558604652, 4.64950704676480919580323590583, 5.19703992661925957481520918925, 5.52344065907571606240257717444, 5.54381359289116914041628820766, 6.15620996062020129395201056662, 6.56558165991719062802854989718, 7.08270744356990329242129282345, 7.08648667788623873489791526150, 7.75274728439861315120651588333, 7.81906273187618179676538576740

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