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Properties

Label 4-4998e2-1.1-c1e2-0-3
Degree $4$
Conductor $24980004$
Sign $1$
Analytic cond. $1592.74$
Root an. cond. $6.31737$
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  − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 4·8-s + 3·9-s + 4·10-s + 4·11-s + 6·12-s − 4·15-s + 5·16-s − 2·17-s − 6·18-s + 6·19-s − 6·20-s − 8·22-s + 6·23-s − 8·24-s − 5·25-s + 4·27-s − 8·29-s + 8·30-s + 12·31-s − 6·32-s + 8·33-s + 4·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s + 1.20·11-s + 1.73·12-s − 1.03·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s + 1.37·19-s − 1.34·20-s − 1.70·22-s + 1.25·23-s − 1.63·24-s − 25-s + 0.769·27-s − 1.48·29-s + 1.46·30-s + 2.15·31-s − 1.06·32-s + 1.39·33-s + 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24980004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24980004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24980004\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1592.74\)
Root analytic conductor: \(6.31737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24980004,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.689330209\)
\(L(\frac12)\) \(\approx\) \(2.689330209\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_j
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_s
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$D_{4}$ \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_cb
29$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_co
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.31.am_du
37$D_{4}$ \( 1 + 6 T + 81 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_dd
41$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_co
43$D_{4}$ \( 1 - 14 T + 127 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_ex
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.47.au_hm
53$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_fe
59$D_{4}$ \( 1 - 10 T + 111 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_eh
61$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_co
67$D_{4}$ \( 1 + 2 T + 103 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_dz
71$D_{4}$ \( 1 - 2 T + 141 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_fl
73$D_{4}$ \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.73.aq_hu
79$D_{4}$ \( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_gg
83$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_fu
89$D_{4}$ \( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.89.k_cx
97$D_{4}$ \( 1 + 8 T + 178 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_gw
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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

−8.381599311768929476024755869488, −8.188088343202154938095708951230, −7.71107271752179426886195045573, −7.47587109513816898410067721729, −7.24327488072229659884759586446, −6.87393977302736831524588310125, −6.46907250804500963390678707509, −6.19236398308072811211386099558, −5.39570881412551241397881051398, −5.35208339480885541967817547688, −4.57161425307637155050470365437, −4.09571230635584524874588463518, −3.78412841667839640162890934921, −3.52598226732704934823301771248, −2.95127468007799322289695926065, −2.59306511475149919301252106104, −2.04160761143890824091897645982, −1.63769594147637879625314852097, −0.845846226692255186696990808460, −0.71415710724654164417893830589, 0.71415710724654164417893830589, 0.845846226692255186696990808460, 1.63769594147637879625314852097, 2.04160761143890824091897645982, 2.59306511475149919301252106104, 2.95127468007799322289695926065, 3.52598226732704934823301771248, 3.78412841667839640162890934921, 4.09571230635584524874588463518, 4.57161425307637155050470365437, 5.35208339480885541967817547688, 5.39570881412551241397881051398, 6.19236398308072811211386099558, 6.46907250804500963390678707509, 6.87393977302736831524588310125, 7.24327488072229659884759586446, 7.47587109513816898410067721729, 7.71107271752179426886195045573, 8.188088343202154938095708951230, 8.381599311768929476024755869488

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