Properties

Label 4-1776e2-1.1-c0e2-0-3
Degree $4$
Conductor $3154176$
Sign $1$
Analytic cond. $0.785597$
Root an. cond. $0.941456$
Motivic weight $0$
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  + 3-s − 7-s + 13-s + 2·19-s − 21-s − 25-s − 27-s + 2·31-s + 2·37-s + 39-s + 2·43-s + 49-s + 2·57-s − 2·61-s − 67-s − 2·73-s − 75-s − 79-s − 81-s − 91-s + 2·93-s − 2·97-s − 4·103-s + 109-s + 2·111-s + 2·121-s + 127-s + ⋯
L(s)  = 1  + 3-s − 7-s + 13-s + 2·19-s − 21-s − 25-s − 27-s + 2·31-s + 2·37-s + 39-s + 2·43-s + 49-s + 2·57-s − 2·61-s − 67-s − 2·73-s − 75-s − 79-s − 81-s − 91-s + 2·93-s − 2·97-s − 4·103-s + 109-s + 2·111-s + 2·121-s + 127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3154176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3154176\)    =    \(2^{8} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.785597\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1776} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3154176,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.606510341\)
\(L(\frac12)\) \(\approx\) \(1.606510341\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2} \)
41$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
73$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−9.677853962000689571077258512864, −9.170672477062316034205995503717, −9.042186843722134281448232793420, −8.559666746269037484949126565118, −8.010529295559822910899513070243, −7.77922638635996294022720573281, −7.49246560253283693970212461781, −7.01151212015942324711392525753, −6.36920046172565192977629468872, −6.12675091943907365783702913901, −5.57084126838916633563934531444, −5.56949854893600664994369288042, −4.42813594897073325934726593499, −4.28065774660506646052382418577, −3.80568205553149369716010253161, −3.05718088586235969492704925960, −2.89681049440655625679880817407, −2.64328669925049778356037693813, −1.59841125263865401988053918924, −0.994463184971430736708565272484, 0.994463184971430736708565272484, 1.59841125263865401988053918924, 2.64328669925049778356037693813, 2.89681049440655625679880817407, 3.05718088586235969492704925960, 3.80568205553149369716010253161, 4.28065774660506646052382418577, 4.42813594897073325934726593499, 5.56949854893600664994369288042, 5.57084126838916633563934531444, 6.12675091943907365783702913901, 6.36920046172565192977629468872, 7.01151212015942324711392525753, 7.49246560253283693970212461781, 7.77922638635996294022720573281, 8.010529295559822910899513070243, 8.559666746269037484949126565118, 9.042186843722134281448232793420, 9.170672477062316034205995503717, 9.677853962000689571077258512864

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