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Properties

Label 4-2448e2-1.1-c1e2-0-35
Degree $4$
Conductor $5992704$
Sign $1$
Analytic cond. $382.100$
Root an. cond. $4.42124$
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  + 5-s + 2·7-s − 9·11-s − 3·13-s + 2·17-s + 3·19-s − 11·23-s − 5·25-s + 6·29-s + 2·35-s − 8·37-s − 7·41-s − 3·43-s − 10·47-s + 6·49-s − 12·53-s − 9·55-s − 2·59-s − 8·61-s − 3·65-s − 24·67-s − 2·71-s − 18·77-s − 4·79-s + 6·83-s + 2·85-s − 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 2.71·11-s − 0.832·13-s + 0.485·17-s + 0.688·19-s − 2.29·23-s − 25-s + 1.11·29-s + 0.338·35-s − 1.31·37-s − 1.09·41-s − 0.457·43-s − 1.45·47-s + 6/7·49-s − 1.64·53-s − 1.21·55-s − 0.260·59-s − 1.02·61-s − 0.372·65-s − 2.93·67-s − 0.237·71-s − 2.05·77-s − 0.450·79-s + 0.658·83-s + 0.216·85-s − 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5992704\)    =    \(2^{8} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(382.100\)
Root analytic conductor: \(4.42124\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5992704,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_g
7$D_{4}$ \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ac
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.11.j_bm
13$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_y
19$D_{4}$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_c
23$D_{4}$ \( 1 + 11 T + 72 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.23.l_cu
29$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_by
31$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.31.a_ag
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$D_{4}$ \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.41.h_ce
43$D_{4}$ \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.43.d_by
47$D_{4}$ \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_dy
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 + 2 T - 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_abi
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$D_{4}$ \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_ak
73$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.73.a_da
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$D_{4}$ \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_gc
89$D_{4}$ \( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.89.k_he
97$D_{4}$ \( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_he
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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

−8.556491516128697851743720547022, −8.273624835617332549233574880453, −7.83157532575406836752797062865, −7.80569646197547009506293796863, −7.42093380760426457102524274879, −6.95443816861363578705607312922, −6.15421566615013254096092063873, −6.10492418686780476505788051907, −5.50261866628589116699448543427, −5.19830291839091449792419528538, −4.84674296579727040910571177274, −4.60564059100914356517463134654, −3.89968005419426666950281285470, −3.26531733917561015548195253751, −2.85209457165435299255999584118, −2.47829054680165211813500209115, −1.69149065130193291566762435180, −1.65781140990140730814880686762, 0, 0, 1.65781140990140730814880686762, 1.69149065130193291566762435180, 2.47829054680165211813500209115, 2.85209457165435299255999584118, 3.26531733917561015548195253751, 3.89968005419426666950281285470, 4.60564059100914356517463134654, 4.84674296579727040910571177274, 5.19830291839091449792419528538, 5.50261866628589116699448543427, 6.10492418686780476505788051907, 6.15421566615013254096092063873, 6.95443816861363578705607312922, 7.42093380760426457102524274879, 7.80569646197547009506293796863, 7.83157532575406836752797062865, 8.273624835617332549233574880453, 8.556491516128697851743720547022

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