Properties

Label 4-903168-1.1-c1e2-0-19
Degree $4$
Conductor $903168$
Sign $-1$
Analytic cond. $57.5867$
Root an. cond. $2.75474$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s − 12·13-s + 4·17-s − 6·25-s + 20·29-s + 20·41-s − 7·49-s − 28·53-s + 20·61-s + 9·81-s + 20·89-s + 36·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 9-s − 3.32·13-s + 0.970·17-s − 6/5·25-s + 3.71·29-s + 3.12·41-s − 49-s − 3.84·53-s + 2.56·61-s + 81-s + 2.11·89-s + 3.32·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 903168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(903168\)    =    \(2^{11} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(57.5867\)
Root analytic conductor: \(2.75474\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{903168} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 903168,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−7.86574055998535746161422058704, −7.77208889793402512778343020047, −7.14549395995132795979264900120, −6.53091467812377191393913765232, −6.27282080646876620831649653268, −5.70026200748580530701756563765, −5.01452297596172644651029655270, −4.87421138112525102501518860592, −4.49261820153658384271905800215, −3.64792213512322410806357507416, −2.88500886362605312928015209245, −2.63895104553179739845027310173, −2.21231952585958536885417842674, −0.980682051952626202561787680881, 0, 0.980682051952626202561787680881, 2.21231952585958536885417842674, 2.63895104553179739845027310173, 2.88500886362605312928015209245, 3.64792213512322410806357507416, 4.49261820153658384271905800215, 4.87421138112525102501518860592, 5.01452297596172644651029655270, 5.70026200748580530701756563765, 6.27282080646876620831649653268, 6.53091467812377191393913765232, 7.14549395995132795979264900120, 7.77208889793402512778343020047, 7.86574055998535746161422058704

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