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Properties

Label 4-8712e2-1.1-c1e2-0-14
Degree $4$
Conductor $75898944$
Sign $1$
Analytic cond. $4839.38$
Root an. cond. $8.34060$
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  − 5-s + 3·7-s + 2·13-s + 8·17-s − 6·19-s − 10·23-s + 2·25-s − 3·29-s − 5·31-s − 3·35-s − 8·37-s − 10·41-s − 6·43-s + 4·49-s − 17·53-s − 13·59-s + 6·61-s − 2·65-s + 16·67-s − 4·71-s − 31·73-s − 3·79-s − 83-s − 8·85-s − 18·89-s + 6·91-s + 6·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s + 0.554·13-s + 1.94·17-s − 1.37·19-s − 2.08·23-s + 2/5·25-s − 0.557·29-s − 0.898·31-s − 0.507·35-s − 1.31·37-s − 1.56·41-s − 0.914·43-s + 4/7·49-s − 2.33·53-s − 1.69·59-s + 0.768·61-s − 0.248·65-s + 1.95·67-s − 0.474·71-s − 3.62·73-s − 0.337·79-s − 0.109·83-s − 0.867·85-s − 1.90·89-s + 0.628·91-s + 0.615·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75898944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75898944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(75898944\)    =    \(2^{6} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4839.38\)
Root analytic conductor: \(8.34060\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 75898944,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
11 \( 1 \)
good5$D_{4}$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ab
7$D_{4}$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_f
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_w
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bq
23$D_{4}$ \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.23.k_co
29$D_{4}$ \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ch
31$D_{4}$ \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_cf
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$D_{4}$ \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_dy
43$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_by
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$D_{4}$ \( 1 + 17 T + 167 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.53.r_gl
59$D_{4}$ \( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.59.n_gd
61$D_{4}$ \( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_g
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_4$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_co
73$D_{4}$ \( 1 + 31 T + 385 T^{2} + 31 p T^{3} + p^{2} T^{4} \) 2.73.bf_ov
79$D_{4}$ \( 1 + 3 T + 159 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.79.d_gd
83$D_{4}$ \( 1 + T + 165 T^{2} + p T^{3} + p^{2} T^{4} \) 2.83.b_gj
89$D_{4}$ \( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_ju
97$D_{4}$ \( 1 + T - 87 T^{2} + p T^{3} + p^{2} T^{4} \) 2.97.b_adj
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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

−7.52152593906657226008472952629, −7.34881631709467685801561008755, −7.11071423775719357510892753740, −6.47773389260812938675478089267, −6.11675198465515062993883087977, −5.95494045571082517165736945930, −5.51935833455287810241479784035, −5.15970776009231410685971735581, −4.76845417582265173654526485958, −4.44205837738084222201093846943, −4.10944145008219791144126591136, −3.57915648653786288194661003015, −3.24195397957570094323919775718, −3.20653773276818271837345545781, −2.12687786634245625724516686065, −1.97992979692738288465172704779, −1.51975917180180027586109261330, −1.19622492791519151853856930398, 0, 0, 1.19622492791519151853856930398, 1.51975917180180027586109261330, 1.97992979692738288465172704779, 2.12687786634245625724516686065, 3.20653773276818271837345545781, 3.24195397957570094323919775718, 3.57915648653786288194661003015, 4.10944145008219791144126591136, 4.44205837738084222201093846943, 4.76845417582265173654526485958, 5.15970776009231410685971735581, 5.51935833455287810241479784035, 5.95494045571082517165736945930, 6.11675198465515062993883087977, 6.47773389260812938675478089267, 7.11071423775719357510892753740, 7.34881631709467685801561008755, 7.52152593906657226008472952629

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