Properties

Label 4-690e2-1.1-c1e2-0-3
Degree $4$
Conductor $476100$
Sign $1$
Analytic cond. $30.3565$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s + 4·10-s + 2·11-s − 6·12-s + 4·13-s − 4·14-s − 4·15-s + 5·16-s + 6·17-s + 6·18-s + 8·19-s + 6·20-s + 4·21-s + 4·22-s + 2·23-s − 8·24-s + 3·25-s + 8·26-s − 4·27-s − 6·28-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 + 0.603·11-s − 1.73·12-s + 1.10·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.45·17-s + 1.41·18-s + 1.83·19-s + 1.34·20-s + 0.872·21-s + 0.852·22-s + 0.417·23-s − 1.63·24-s + 3/5·25-s + 1.56·26-s − 0.769·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(476100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(30.3565\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{690} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 476100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.617676150\)
\(L(\frac12)\) \(\approx\) \(4.617676150\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
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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

−10.70347967111563947986137354215, −10.38715393270090526032149908101, −9.893093454040967770924047872289, −9.727850804397065485484578916840, −9.079674642465288933737635760686, −8.600193152708372041192707561691, −7.72790984760758369525384065297, −7.47348668314169995430620001292, −6.80283966174642378824177880289, −6.46063291197120849703060187639, −6.19114578083713139793062404786, −5.66120849697089272939956247000, −5.18447987062167723771357045743, −5.16330293180140880866797513771, −4.21035356013394717146926553200, −3.70711499408325527712209289758, −3.19008979336382757358983849240, −2.73794016074583526324836268727, −1.41641795769549472108387942963, −1.22689005674514361395070258668, 1.22689005674514361395070258668, 1.41641795769549472108387942963, 2.73794016074583526324836268727, 3.19008979336382757358983849240, 3.70711499408325527712209289758, 4.21035356013394717146926553200, 5.16330293180140880866797513771, 5.18447987062167723771357045743, 5.66120849697089272939956247000, 6.19114578083713139793062404786, 6.46063291197120849703060187639, 6.80283966174642378824177880289, 7.47348668314169995430620001292, 7.72790984760758369525384065297, 8.600193152708372041192707561691, 9.079674642465288933737635760686, 9.727850804397065485484578916840, 9.893093454040967770924047872289, 10.38715393270090526032149908101, 10.70347967111563947986137354215

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