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Properties

Label 4-4032e2-1.1-c1e2-0-25
Degree $4$
Conductor $16257024$
Sign $1$
Analytic cond. $1036.56$
Root an. cond. $5.67412$
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  − 2·7-s + 4·11-s − 4·13-s − 8·17-s − 4·23-s + 2·25-s + 4·29-s − 8·31-s + 4·37-s − 16·41-s + 16·43-s − 8·47-s + 3·49-s − 4·53-s − 16·59-s + 4·61-s − 8·67-s − 12·71-s + 12·73-s − 8·77-s − 8·79-s − 8·83-s + 8·91-s − 4·97-s − 24·101-s − 8·103-s + 4·107-s + ⋯
L(s)  = 1  − 0.755·7-s + 1.20·11-s − 1.10·13-s − 1.94·17-s − 0.834·23-s + 2/5·25-s + 0.742·29-s − 1.43·31-s + 0.657·37-s − 2.49·41-s + 2.43·43-s − 1.16·47-s + 3/7·49-s − 0.549·53-s − 2.08·59-s + 0.512·61-s − 0.977·67-s − 1.42·71-s + 1.40·73-s − 0.911·77-s − 0.900·79-s − 0.878·83-s + 0.838·91-s − 0.406·97-s − 2.38·101-s − 0.788·103-s + 0.386·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16257024\)    =    \(2^{12} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1036.56\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 16257024,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_o
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bm
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_bm
29$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_o
31$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_be
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$D_{4}$ \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.41.q_fe
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_ck
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$D_{4}$ \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_fe
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_da
67$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_dy
71$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_gk
73$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_fe
79$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ew
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \) 2.89.a_gk
97$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_fu
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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

−8.195438326036946941655415211622, −7.992846356885766983001991000556, −7.37136082751707015251643770465, −7.12501562426161995283832293308, −6.67982625313640665743777995138, −6.61355207700725938194063597725, −6.09965788745713556658947745090, −5.75314205747752928299971741415, −5.31828070953430080844729252017, −4.67967400312300377332191244689, −4.35540955853370115709450187940, −4.30895469453670692609610459919, −3.53038146336328772307829236318, −3.26225690426445148421076976505, −2.66783289303193059459661940737, −2.27057239513044932154201661692, −1.73507727708640153423590662462, −1.21568558898854458089765081722, 0, 0, 1.21568558898854458089765081722, 1.73507727708640153423590662462, 2.27057239513044932154201661692, 2.66783289303193059459661940737, 3.26225690426445148421076976505, 3.53038146336328772307829236318, 4.30895469453670692609610459919, 4.35540955853370115709450187940, 4.67967400312300377332191244689, 5.31828070953430080844729252017, 5.75314205747752928299971741415, 6.09965788745713556658947745090, 6.61355207700725938194063597725, 6.67982625313640665743777995138, 7.12501562426161995283832293308, 7.37136082751707015251643770465, 7.992846356885766983001991000556, 8.195438326036946941655415211622

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