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Properties

Label 4-6384e2-1.1-c1e2-0-9
Degree $4$
Conductor $40755456$
Sign $1$
Analytic cond. $2598.60$
Root an. cond. $7.13978$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·7-s + 3·9-s − 4·11-s − 2·19-s − 4·21-s − 4·23-s − 2·25-s + 4·27-s + 4·29-s − 8·31-s − 8·33-s + 4·37-s − 4·41-s − 16·43-s + 3·49-s + 4·53-s − 4·57-s + 8·59-s − 4·61-s − 6·63-s − 20·67-s − 8·69-s + 8·71-s − 12·73-s − 4·75-s + 8·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 9-s − 1.20·11-s − 0.458·19-s − 0.872·21-s − 0.834·23-s − 2/5·25-s + 0.769·27-s + 0.742·29-s − 1.43·31-s − 1.39·33-s + 0.657·37-s − 0.624·41-s − 2.43·43-s + 3/7·49-s + 0.549·53-s − 0.529·57-s + 1.04·59-s − 0.512·61-s − 0.755·63-s − 2.44·67-s − 0.963·69-s + 0.949·71-s − 1.40·73-s − 0.461·75-s + 0.911·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40755456\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(2598.60\)
Root analytic conductor: \(7.13978\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40755456,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_bq
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_be
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cs
37$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_g
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cc
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.47.a_di
53$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_da
59$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_dy
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_ac
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ew
73$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_fu
79$D_{4}$ \( 1 - 12 T + 186 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_he
83$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.83.a_bm
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_de
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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

−7.86936009335166280823836277351, −7.66680224764490507555531813850, −7.10168349987249283410269549321, −6.96713868826347347861929559649, −6.41345368500381701368320915916, −6.28300535241647708406574508837, −5.52939615381435575439154500035, −5.49782875112792288550556337822, −4.96820009889317128377389393233, −4.52317245590552112360044059549, −3.98371667735616950321781318018, −3.90136767644913125721922483917, −3.22037150882976994374167045890, −3.05744508154651150166606782529, −2.56069661635253052208610395959, −2.22001250128405304542165422325, −1.69055406868545000811437621423, −1.23156185915157281683590494997, 0, 0, 1.23156185915157281683590494997, 1.69055406868545000811437621423, 2.22001250128405304542165422325, 2.56069661635253052208610395959, 3.05744508154651150166606782529, 3.22037150882976994374167045890, 3.90136767644913125721922483917, 3.98371667735616950321781318018, 4.52317245590552112360044059549, 4.96820009889317128377389393233, 5.49782875112792288550556337822, 5.52939615381435575439154500035, 6.28300535241647708406574508837, 6.41345368500381701368320915916, 6.96713868826347347861929559649, 7.10168349987249283410269549321, 7.66680224764490507555531813850, 7.86936009335166280823836277351

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