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Properties

Label 4-6e8-1.1-c1e2-0-37
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
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 − 3·7-s − 2·11-s + 5·13-s − 5·17-s − 7·19-s − 5·23-s − 25-s − 3·29-s − 7·31-s + 3·35-s + 6·37-s − 12·41-s − 8·43-s − 3·47-s + 49-s − 10·53-s + 2·55-s − 14·59-s − 61-s − 5·65-s − 4·67-s + 8·71-s − 7·73-s + 6·77-s + 7·79-s − 25·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s − 0.603·11-s + 1.38·13-s − 1.21·17-s − 1.60·19-s − 1.04·23-s − 1/5·25-s − 0.557·29-s − 1.25·31-s + 0.507·35-s + 0.986·37-s − 1.87·41-s − 1.21·43-s − 0.437·47-s + 1/7·49-s − 1.37·53-s + 0.269·55-s − 1.82·59-s − 0.128·61-s − 0.620·65-s − 0.488·67-s + 0.949·71-s − 0.819·73-s + 0.683·77-s + 0.787·79-s − 2.74·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_c
7$D_{4}$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_i
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$D_{4}$ \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_y
17$D_{4}$ \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.17.f_bg
19$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.19.h_bq
23$D_{4}$ \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bs
29$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ca
31$D_{4}$ \( 1 + 7 T + 66 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.31.h_co
37$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_by
41$D_{4}$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dh
43$D_{4}$ \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cr
47$D_{4}$ \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.47.d_dk
53$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.53.k_du
59$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.59.o_gl
61$D_{4}$ \( 1 + T + 114 T^{2} + p T^{3} + p^{2} T^{4} \) 2.61.b_ek
67$D_{4}$ \( 1 + 4 T + 105 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_eb
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$D_{4}$ \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.73.h_dg
79$D_{4}$ \( 1 - 7 T + 162 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.79.ah_gg
83$D_{4}$ \( 1 + 25 T + 314 T^{2} + 25 p T^{3} + p^{2} T^{4} \) 2.83.z_mc
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$D_{4}$ \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_gv
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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

−9.394158874294007273996782172605, −8.979472142573530964819046455050, −8.727770223738188031122951495364, −8.121269556712013142674166634499, −8.041178318912997030205839562982, −7.44786911605943622831691737688, −6.80774154888078831344379910125, −6.56769978400851266016617686027, −6.08502863571544441518884100940, −6.01212845975460546969550924587, −5.17202413054534468385232123754, −4.74568525600728611458822717237, −4.04015392546546353758577951625, −3.92564790152860563571748785695, −3.24762037428474586875954999195, −2.89236116533902573643406665998, −1.94950947461035593346148136329, −1.67130554709750835854018785293, 0, 0, 1.67130554709750835854018785293, 1.94950947461035593346148136329, 2.89236116533902573643406665998, 3.24762037428474586875954999195, 3.92564790152860563571748785695, 4.04015392546546353758577951625, 4.74568525600728611458822717237, 5.17202413054534468385232123754, 6.01212845975460546969550924587, 6.08502863571544441518884100940, 6.56769978400851266016617686027, 6.80774154888078831344379910125, 7.44786911605943622831691737688, 8.041178318912997030205839562982, 8.121269556712013142674166634499, 8.727770223738188031122951495364, 8.979472142573530964819046455050, 9.394158874294007273996782172605

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