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Properties

Label 4-1998e2-1.1-c1e2-0-6
Degree $4$
Conductor $3992004$
Sign $1$
Analytic cond. $254.533$
Root an. cond. $3.99425$
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-s − 4·5-s + 8-s + 4·10-s − 11-s − 6·13-s − 16-s − 6·17-s − 2·19-s + 22-s − 6·23-s + 5·25-s + 6·26-s + 2·31-s + 6·34-s + 2·37-s + 2·38-s − 4·40-s + 9·41-s + 5·43-s + 6·46-s − 4·47-s + 7·49-s − 5·50-s + 24·53-s + 4·55-s + 11·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.78·5-s + 0.353·8-s + 1.26·10-s − 0.301·11-s − 1.66·13-s − 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.213·22-s − 1.25·23-s + 25-s + 1.17·26-s + 0.359·31-s + 1.02·34-s + 0.328·37-s + 0.324·38-s − 0.632·40-s + 1.40·41-s + 0.762·43-s + 0.884·46-s − 0.583·47-s + 49-s − 0.707·50-s + 3.29·53-s + 0.539·55-s + 1.43·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3992004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3992004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3992004\)    =    \(2^{2} \cdot 3^{6} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(254.533\)
Root analytic conductor: \(3.99425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3992004,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_ak
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_abb
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.af_as
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_abf
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$C_2^2$ \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.59.al_ck
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.67.p_gc
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.73.k_gp
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_abr
83$C_2^2$ \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_acp
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 3 T - 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.97.d_adk
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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

−8.928686424717250775149414870933, −8.649146624575688245524746059323, −8.065186816936693194567145490822, −7.84041142020347001886168494097, −7.51853038876302825080820898129, −7.15715247819730376089081580720, −6.92793515304689193578116098461, −6.29010850269227281604307441624, −5.75795471539523000290161488783, −5.36975184231244396303259709521, −4.69655188706862126131138542089, −4.30987491784774093096540427448, −4.07135256714463798555079054423, −3.84487994009819231013128912384, −2.75409663861404236570072778770, −2.56598614523672952481243465395, −2.04484762341683618216811772710, −0.991381727047543434765360080136, 0, 0, 0.991381727047543434765360080136, 2.04484762341683618216811772710, 2.56598614523672952481243465395, 2.75409663861404236570072778770, 3.84487994009819231013128912384, 4.07135256714463798555079054423, 4.30987491784774093096540427448, 4.69655188706862126131138542089, 5.36975184231244396303259709521, 5.75795471539523000290161488783, 6.29010850269227281604307441624, 6.92793515304689193578116098461, 7.15715247819730376089081580720, 7.51853038876302825080820898129, 7.84041142020347001886168494097, 8.065186816936693194567145490822, 8.649146624575688245524746059323, 8.928686424717250775149414870933

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