Properties

Label 4-1494108-1.1-c1e2-0-0
Degree $4$
Conductor $1494108$
Sign $1$
Analytic cond. $95.2656$
Root an. cond. $3.12416$
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 + 3·4-s − 7-s − 4·8-s + 9-s − 2·11-s + 2·14-s + 5·16-s − 2·18-s + 4·22-s − 8·23-s − 10·25-s − 3·28-s + 12·29-s − 6·32-s + 3·36-s + 20·37-s + 16·43-s − 6·44-s + 16·46-s + 49-s + 20·50-s − 20·53-s + 4·56-s − 24·58-s − 63-s + 7·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 1/3·9-s − 0.603·11-s + 0.534·14-s + 5/4·16-s − 0.471·18-s + 0.852·22-s − 1.66·23-s − 2·25-s − 0.566·28-s + 2.22·29-s − 1.06·32-s + 1/2·36-s + 3.28·37-s + 2.43·43-s − 0.904·44-s + 2.35·46-s + 1/7·49-s + 2.82·50-s − 2.74·53-s + 0.534·56-s − 3.15·58-s − 0.125·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1494108\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(95.2656\)
Root analytic conductor: \(3.12416\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1494108} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1494108,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7197813021\)
\(L(\frac12)\) \(\approx\) \(0.7197813021\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−7.83922398202642458587198828500, −7.79089756844307556875120056782, −7.30932067156140291452614810498, −6.50358738357793944503631024403, −6.37119763678498512356952453412, −5.90660738221292552490908354487, −5.61053425475447892671224791397, −4.70213202876017617150319733849, −4.23003763111100218531402281021, −3.88277706681863956744577131696, −2.90909027996993607554406630510, −2.63621530561166833588250288036, −2.07103752121128576013331888677, −1.28532701621031142017445445957, −0.48386265623252023695541121788, 0.48386265623252023695541121788, 1.28532701621031142017445445957, 2.07103752121128576013331888677, 2.63621530561166833588250288036, 2.90909027996993607554406630510, 3.88277706681863956744577131696, 4.23003763111100218531402281021, 4.70213202876017617150319733849, 5.61053425475447892671224791397, 5.90660738221292552490908354487, 6.37119763678498512356952453412, 6.50358738357793944503631024403, 7.30932067156140291452614810498, 7.79089756844307556875120056782, 7.83922398202642458587198828500

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