Properties

Label 4-1134e2-1.1-c1e2-0-22
Degree $4$
Conductor $1285956$
Sign $1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
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 − 5-s − 5·7-s + 4·8-s − 2·10-s − 5·11-s − 10·14-s + 5·16-s + 4·17-s − 8·19-s − 3·20-s − 10·22-s + 4·23-s + 5·25-s − 15·28-s + 5·29-s + 6·31-s + 6·32-s + 8·34-s + 5·35-s + 4·37-s − 16·38-s − 4·40-s − 2·43-s − 15·44-s + 8·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.447·5-s − 1.88·7-s + 1.41·8-s − 0.632·10-s − 1.50·11-s − 2.67·14-s + 5/4·16-s + 0.970·17-s − 1.83·19-s − 0.670·20-s − 2.13·22-s + 0.834·23-s + 25-s − 2.83·28-s + 0.928·29-s + 1.07·31-s + 1.06·32-s + 1.37·34-s + 0.845·35-s + 0.657·37-s − 2.59·38-s − 0.632·40-s − 0.304·43-s − 2.26·44-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1134} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.599010994\)
\(L(\frac12)\) \(\approx\) \(2.599010994\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7 T - 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−10.24325568620545621264617678335, −9.722815032513935836087037178044, −9.290587224480628138235668515716, −8.748594456877857504775649499084, −8.246954869797783614569292344123, −7.82696881808736457327883071137, −7.50963196714324918913973012470, −6.82963748780639103006076793117, −6.50085396884058343081308401086, −6.35268468604198654364906778511, −5.78913350808345790580688620004, −5.27967609763227623772526903188, −4.81077483172443124844892896363, −4.37094120031666981405274753454, −3.94943249384637026847873623693, −3.08468524151635009491501527226, −2.93240083776323606761493869618, −2.79876466069493092936522099232, −1.73792528971302509004567208480, −0.53574952050694922372747049499, 0.53574952050694922372747049499, 1.73792528971302509004567208480, 2.79876466069493092936522099232, 2.93240083776323606761493869618, 3.08468524151635009491501527226, 3.94943249384637026847873623693, 4.37094120031666981405274753454, 4.81077483172443124844892896363, 5.27967609763227623772526903188, 5.78913350808345790580688620004, 6.35268468604198654364906778511, 6.50085396884058343081308401086, 6.82963748780639103006076793117, 7.50963196714324918913973012470, 7.82696881808736457327883071137, 8.246954869797783614569292344123, 8.748594456877857504775649499084, 9.290587224480628138235668515716, 9.722815032513935836087037178044, 10.24325568620545621264617678335

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