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Properties

Label 4-1368e2-1.1-c1e2-0-23
Degree $4$
Conductor $1871424$
Sign $1$
Analytic cond. $119.323$
Root an. cond. $3.30507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 6·11-s − 2·13-s + 2·17-s + 19-s + 6·23-s + 5·25-s − 4·29-s − 20·31-s + 4·37-s + 9·41-s + 4·43-s − 12·47-s − 14·49-s − 2·53-s + 24·55-s − 59-s + 8·61-s + 8·65-s − 9·67-s − 6·71-s + 9·73-s + 4·79-s + 10·83-s − 8·85-s − 18·89-s − 4·95-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.80·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 25-s − 0.742·29-s − 3.59·31-s + 0.657·37-s + 1.40·41-s + 0.609·43-s − 1.75·47-s − 2·49-s − 0.274·53-s + 3.23·55-s − 0.130·59-s + 1.02·61-s + 0.992·65-s − 1.09·67-s − 0.712·71-s + 1.05·73-s + 0.450·79-s + 1.09·83-s − 0.867·85-s − 1.90·89-s − 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1871424\)    =    \(2^{6} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(119.323\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1871424,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
19$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_an
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.31.u_gg
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dt
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_abx
59$C_2^2$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) 2.59.b_acg
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_d
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_o
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.73.aj_i
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.83.ak_hj
89$C_2^2$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_jb
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) 2.97.b_ads
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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

−9.457994125107719805097877520386, −9.107795273162901345864065081160, −8.216124615608848005661779157040, −8.200689517438082779594401408425, −7.72290427755242095321194105477, −7.45095371135005017210275298730, −7.22401073194221916980695950331, −6.71913611105134990652463654128, −5.98152341278566681831237526760, −5.43214842127881589586722106763, −5.20209298951459580357708473155, −4.81663906891376778980749438302, −4.11240635181886462407126773576, −3.78285456924514605925214456129, −3.24943920395507697830303122771, −2.85589525382821161172138339512, −2.19113075987656725040316578571, −1.34720092598821044414781378375, 0, 0, 1.34720092598821044414781378375, 2.19113075987656725040316578571, 2.85589525382821161172138339512, 3.24943920395507697830303122771, 3.78285456924514605925214456129, 4.11240635181886462407126773576, 4.81663906891376778980749438302, 5.20209298951459580357708473155, 5.43214842127881589586722106763, 5.98152341278566681831237526760, 6.71913611105134990652463654128, 7.22401073194221916980695950331, 7.45095371135005017210275298730, 7.72290427755242095321194105477, 8.200689517438082779594401408425, 8.216124615608848005661779157040, 9.107795273162901345864065081160, 9.457994125107719805097877520386

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