Properties

Label 4-1932e2-1.1-c1e2-0-7
Degree $4$
Conductor $3732624$
Sign $1$
Analytic cond. $237.995$
Root an. cond. $3.92773$
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  − 2·3-s + 5-s + 2·7-s + 3·9-s − 5·13-s − 2·15-s − 6·17-s − 8·19-s − 4·21-s − 2·23-s − 6·25-s − 4·27-s + 2·29-s + 2·31-s + 2·35-s − 2·37-s + 10·39-s − 4·41-s − 5·43-s + 3·45-s − 14·47-s + 3·49-s + 12·51-s + 15·53-s + 16·57-s + 11·59-s − 61-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 0.755·7-s + 9-s − 1.38·13-s − 0.516·15-s − 1.45·17-s − 1.83·19-s − 0.872·21-s − 0.417·23-s − 6/5·25-s − 0.769·27-s + 0.371·29-s + 0.359·31-s + 0.338·35-s − 0.328·37-s + 1.60·39-s − 0.624·41-s − 0.762·43-s + 0.447·45-s − 2.04·47-s + 3/7·49-s + 1.68·51-s + 2.06·53-s + 2.11·57-s + 1.43·59-s − 0.128·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3732624\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(237.995\)
Root analytic conductor: \(3.92773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1932} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3732624,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 5 T + 29 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 23 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 15 T + 133 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 11 T + 119 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 + T + 41 T^{2} + p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 5 T - 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 13 T + 155 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 245 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 3 T + 177 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
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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

−8.865573702542904813041700199964, −8.651362878125213709904818829322, −8.232838705689800509783592630073, −7.87823164910431246022748568392, −7.25276574617372903731106419117, −6.95438563857432411132635324615, −6.46407624520877745263977758138, −6.44940168022507556725179244512, −5.62332677581800205276268885908, −5.45265865247141905259912597397, −5.01006435552063346040420928873, −4.49217842734020754909329312001, −4.16807982310039301498395542834, −3.95298508279997642186992116669, −2.76289467099899513060619439256, −2.46817203600319115178278712920, −1.78986418343998711783463871063, −1.50373751912312532066113730990, 0, 0, 1.50373751912312532066113730990, 1.78986418343998711783463871063, 2.46817203600319115178278712920, 2.76289467099899513060619439256, 3.95298508279997642186992116669, 4.16807982310039301498395542834, 4.49217842734020754909329312001, 5.01006435552063346040420928873, 5.45265865247141905259912597397, 5.62332677581800205276268885908, 6.44940168022507556725179244512, 6.46407624520877745263977758138, 6.95438563857432411132635324615, 7.25276574617372903731106419117, 7.87823164910431246022748568392, 8.232838705689800509783592630073, 8.651362878125213709904818829322, 8.865573702542904813041700199964

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