Properties

Label 4-901404-1.1-c1e2-0-0
Degree $4$
Conductor $901404$
Sign $1$
Analytic cond. $57.4743$
Root an. cond. $2.75339$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s − 7-s + 9-s + 2·12-s + 2·13-s + 16-s + 2·21-s + 2·25-s + 4·27-s + 28-s + 8·31-s − 36-s + 16·37-s − 4·39-s − 14·43-s − 2·48-s + 49-s − 2·52-s − 12·61-s − 63-s − 64-s + 73-s − 4·75-s − 4·79-s − 11·81-s − 2·84-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s + 1/4·16-s + 0.436·21-s + 2/5·25-s + 0.769·27-s + 0.188·28-s + 1.43·31-s − 1/6·36-s + 2.63·37-s − 0.640·39-s − 2.13·43-s − 0.288·48-s + 1/7·49-s − 0.277·52-s − 1.53·61-s − 0.125·63-s − 1/8·64-s + 0.117·73-s − 0.461·75-s − 0.450·79-s − 1.22·81-s − 0.218·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(901404\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 73\)
Sign: $1$
Analytic conductor: \(57.4743\)
Root analytic conductor: \(2.75339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{901404} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9379004215\)
\(L(\frac12)\) \(\approx\) \(0.9379004215\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_1$ \( 1 + T \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.206327212375511730844697930987, −7.77182889185260405621520378174, −7.22417146473227191200512150183, −6.68091203190693064614394892984, −6.23460302623036220826637152780, −6.05478025149282823923822026226, −5.58192981977071434419273498702, −4.90783334524636256510595043651, −4.59016733721661353006644054030, −4.26314361653459966825747525633, −3.31275994520441233914210212452, −3.09685308937199660948416070339, −2.22078472910662239456073336620, −1.21929498383852468040574999783, −0.56350505744805430499663274302, 0.56350505744805430499663274302, 1.21929498383852468040574999783, 2.22078472910662239456073336620, 3.09685308937199660948416070339, 3.31275994520441233914210212452, 4.26314361653459966825747525633, 4.59016733721661353006644054030, 4.90783334524636256510595043651, 5.58192981977071434419273498702, 6.05478025149282823923822026226, 6.23460302623036220826637152780, 6.68091203190693064614394892984, 7.22417146473227191200512150183, 7.77182889185260405621520378174, 8.206327212375511730844697930987

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