Properties

Label 4-390e2-1.1-c1e2-0-1
Degree $4$
Conductor $152100$
Sign $1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 2·5-s − 6-s − 2·7-s − 8-s − 2·10-s + 3·11-s + 2·13-s − 2·14-s + 2·15-s − 16-s − 6·17-s − 2·19-s + 2·21-s + 3·22-s − 3·23-s + 24-s + 3·25-s + 2·26-s + 27-s − 3·29-s + 2·30-s + 10·31-s − 3·33-s − 6·34-s + 4·35-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s − 0.632·10-s + 0.904·11-s + 0.554·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.436·21-s + 0.639·22-s − 0.625·23-s + 0.204·24-s + 3/5·25-s + 0.392·26-s + 0.192·27-s − 0.557·29-s + 0.365·30-s + 1.79·31-s − 0.522·33-s − 1.02·34-s + 0.676·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{390} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.076861218\)
\(L(\frac12)\) \(\approx\) \(1.076861218\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−11.96944610607688778078428759617, −11.18470672148175451806721444570, −10.90991853717775339703645437444, −10.27225866606502018973349429060, −9.622572204555609002449250470717, −9.414998819847251517887214349947, −8.635148008272364352953788854516, −8.422951998527558142078668580528, −7.84728978776286866496109652816, −7.12876347208133567332506338069, −6.53013638269595034694844747068, −6.31237324362766398925068540961, −6.03193150472206187242001455192, −4.89815927260507181081287632967, −4.79401916991276547970907600770, −3.87027621141971682060981747355, −3.82812242145949558112533456870, −2.95818843219766714789245127920, −2.07628193268206550603131263378, −0.63982262903304676713432413512, 0.63982262903304676713432413512, 2.07628193268206550603131263378, 2.95818843219766714789245127920, 3.82812242145949558112533456870, 3.87027621141971682060981747355, 4.79401916991276547970907600770, 4.89815927260507181081287632967, 6.03193150472206187242001455192, 6.31237324362766398925068540961, 6.53013638269595034694844747068, 7.12876347208133567332506338069, 7.84728978776286866496109652816, 8.422951998527558142078668580528, 8.635148008272364352953788854516, 9.414998819847251517887214349947, 9.622572204555609002449250470717, 10.27225866606502018973349429060, 10.90991853717775339703645437444, 11.18470672148175451806721444570, 11.96944610607688778078428759617

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