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Properties

Label 4-5472e2-1.1-c1e2-0-11
Degree $4$
Conductor $29942784$
Sign $1$
Analytic cond. $1909.17$
Root an. cond. $6.61015$
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 − 6·7-s − 3·11-s + 13-s + 8·17-s − 2·19-s + 7·23-s − 5·25-s + 3·29-s − 10·31-s − 6·35-s + 6·37-s − 8·41-s − 15·43-s + 5·47-s + 13·49-s + 13·53-s − 3·55-s − 13·59-s + 9·61-s + 65-s + 5·67-s + 8·71-s − 2·73-s + 18·77-s − 20·79-s − 10·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 2.26·7-s − 0.904·11-s + 0.277·13-s + 1.94·17-s − 0.458·19-s + 1.45·23-s − 25-s + 0.557·29-s − 1.79·31-s − 1.01·35-s + 0.986·37-s − 1.24·41-s − 2.28·43-s + 0.729·47-s + 13/7·49-s + 1.78·53-s − 0.404·55-s − 1.69·59-s + 1.15·61-s + 0.124·65-s + 0.610·67-s + 0.949·71-s − 0.234·73-s + 2.05·77-s − 2.25·79-s − 1.09·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29942784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29942784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29942784\)    =    \(2^{10} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1909.17\)
Root analytic conductor: \(6.61015\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29942784,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_g
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$D_{4}$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_u
13$D_{4}$ \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_w
17$D_{4}$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bh
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.23.ah_cc
29$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_w
31$D_{4}$ \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_cs
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_co
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.41.i_du
43$D_{4}$ \( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.43.p_fi
47$D_{4}$ \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.47.af_ck
53$D_{4}$ \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.53.an_fo
59$D_{4}$ \( 1 + 13 T + 156 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.59.n_ga
61$D_{4}$ \( 1 - 9 T + 36 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_bk
67$D_{4}$ \( 1 - 5 T + 136 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.67.af_fg
71$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_dm
73$D_{4}$ \( 1 + 2 T + 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_db
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$D_{4}$ \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_bm
89$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_bi
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_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

−7.931571899197010201186477161370, −7.65934102151020780431021102657, −7.12149958976135936303684944097, −6.94854282590680218561101981731, −6.40175927427797510643681330888, −6.39539631079424152812857613338, −5.66451389802864086930855511306, −5.64294379198195820592619925374, −5.11274588711729979301832759421, −4.98199727832466716107827077749, −4.00136847697353934091396102154, −3.84435845950123108801806292523, −3.34224313025228774535505110256, −3.17176707117632903226314456893, −2.61706571342565219350796075018, −2.38081762399272265330846658772, −1.45821400708982623155694557100, −1.14884918448503156209941236285, 0, 0, 1.14884918448503156209941236285, 1.45821400708982623155694557100, 2.38081762399272265330846658772, 2.61706571342565219350796075018, 3.17176707117632903226314456893, 3.34224313025228774535505110256, 3.84435845950123108801806292523, 4.00136847697353934091396102154, 4.98199727832466716107827077749, 5.11274588711729979301832759421, 5.64294379198195820592619925374, 5.66451389802864086930855511306, 6.39539631079424152812857613338, 6.40175927427797510643681330888, 6.94854282590680218561101981731, 7.12149958976135936303684944097, 7.65934102151020780431021102657, 7.931571899197010201186477161370

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