Properties

Label 4-1494108-1.1-c1e2-0-16
Degree $4$
Conductor $1494108$
Sign $-1$
Analytic cond. $95.2656$
Root an. cond. $3.12416$
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  − 2·2-s + 3·4-s + 7-s − 4·8-s + 9-s + 2·11-s − 2·14-s + 5·16-s − 2·18-s − 4·22-s − 8·23-s − 6·25-s + 3·28-s + 4·29-s − 6·32-s + 3·36-s − 4·37-s + 6·44-s + 16·46-s + 49-s + 12·50-s − 28·53-s − 4·56-s − 8·58-s + 63-s + 7·64-s + 8·67-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s + 1/3·9-s + 0.603·11-s − 0.534·14-s + 5/4·16-s − 0.471·18-s − 0.852·22-s − 1.66·23-s − 6/5·25-s + 0.566·28-s + 0.742·29-s − 1.06·32-s + 1/2·36-s − 0.657·37-s + 0.904·44-s + 2.35·46-s + 1/7·49-s + 1.69·50-s − 3.84·53-s − 0.534·56-s − 1.05·58-s + 0.125·63-s + 7/8·64-s + 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1494108\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(95.2656\)
Root analytic conductor: \(3.12416\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1494108} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1494108,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.84439006039971069748622971041, −7.49024027339887201227457052610, −6.91394332771191655575545668718, −6.43679449975526542026435018487, −6.17569848665337782312463950999, −5.79323544635467042022952173625, −4.93054069578340184917711747999, −4.72832473817225769292057457440, −3.72716170584453900058590617219, −3.71446826599373971796272627437, −2.83711647406054313257956134718, −1.96943863395594925899870544120, −1.87596574858564202101556695192, −1.00454239585262764496288710387, 0, 1.00454239585262764496288710387, 1.87596574858564202101556695192, 1.96943863395594925899870544120, 2.83711647406054313257956134718, 3.71446826599373971796272627437, 3.72716170584453900058590617219, 4.72832473817225769292057457440, 4.93054069578340184917711747999, 5.79323544635467042022952173625, 6.17569848665337782312463950999, 6.43679449975526542026435018487, 6.91394332771191655575545668718, 7.49024027339887201227457052610, 7.84439006039971069748622971041

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