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Properties

Label 4-6e8-1.1-c1e2-0-13
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 $0$

Origins

Origins of factors

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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: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.381781691\)
\(L(\frac12)\) \(\approx\) \(2.381781691\)
\(L(\frac{3}{2})\) 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.ab_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.ac_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.af_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.af_bs
29$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_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.am_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.ad_dk
53$D_{4}$ \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.53.ak_du
59$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.59.ao_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.i_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.az_mc
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_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

−9.631776519077411429459061501138, −9.612987703939907519702751593101, −8.965702564228445759132617885597, −8.797566020418974118606375944347, −8.360217760389229813738921635692, −7.86370042178954888050792310545, −7.30966098815369833582236284677, −6.94369559749995776772373111889, −6.36121670194464273730365479425, −6.29680116914789375726922615927, −5.62736689584681995744058291532, −5.58045729534457389622656875023, −4.65669765388511646440818917173, −4.22633217333827336929026314240, −3.58184897069210958814638415701, −3.48806856242319950986842161860, −2.70261549260161757630891647851, −2.17282584504374186561074548539, −1.34281739463773241963802371890, −0.70592632603964084143015223490, 0.70592632603964084143015223490, 1.34281739463773241963802371890, 2.17282584504374186561074548539, 2.70261549260161757630891647851, 3.48806856242319950986842161860, 3.58184897069210958814638415701, 4.22633217333827336929026314240, 4.65669765388511646440818917173, 5.58045729534457389622656875023, 5.62736689584681995744058291532, 6.29680116914789375726922615927, 6.36121670194464273730365479425, 6.94369559749995776772373111889, 7.30966098815369833582236284677, 7.86370042178954888050792310545, 8.360217760389229813738921635692, 8.797566020418974118606375944347, 8.965702564228445759132617885597, 9.612987703939907519702751593101, 9.631776519077411429459061501138

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