Properties

Label 4-901404-1.1-c1e2-0-8
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 7-s + 9-s + 2·12-s − 10·13-s + 16-s − 8·19-s − 2·21-s − 6·25-s + 4·27-s − 28-s − 36-s + 20·39-s − 6·43-s − 2·48-s + 49-s + 10·52-s + 16·57-s − 4·61-s + 63-s − 64-s − 73-s + 12·75-s + 8·76-s − 4·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 0.377·7-s + 1/3·9-s + 0.577·12-s − 2.77·13-s + 1/4·16-s − 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.769·27-s − 0.188·28-s − 1/6·36-s + 3.20·39-s − 0.914·43-s − 0.288·48-s + 1/7·49-s + 1.38·52-s + 2.11·57-s − 0.512·61-s + 0.125·63-s − 1/8·64-s − 0.117·73-s + 1.38·75-s + 0.917·76-s − 0.450·79-s − 1.22·81-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: \(2\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.66741287713101616060491401220, −7.27453250538357652270056875157, −6.80202421361833053915690823863, −6.41723488963056388740795526023, −5.82209492493951250848968703517, −5.44714138224514291218356175490, −4.96255306008123813481614349787, −4.62587775930648764410747811531, −4.27920099633952313667377461240, −3.60950273850585120171259945794, −2.60917711246969782569250510083, −2.35003418223929968558351322177, −1.46810796032152333525902376304, 0, 0, 1.46810796032152333525902376304, 2.35003418223929968558351322177, 2.60917711246969782569250510083, 3.60950273850585120171259945794, 4.27920099633952313667377461240, 4.62587775930648764410747811531, 4.96255306008123813481614349787, 5.44714138224514291218356175490, 5.82209492493951250848968703517, 6.41723488963056388740795526023, 6.80202421361833053915690823863, 7.27453250538357652270056875157, 7.66741287713101616060491401220

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