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Properties

Label 4-4998e2-1.1-c1e2-0-12
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 $2$

Origins

Origins of factors

Downloads

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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 − 6·11-s − 6·12-s − 2·13-s − 4·15-s + 5·16-s + 2·17-s + 6·18-s + 4·19-s + 6·20-s − 12·22-s − 12·23-s − 8·24-s − 7·25-s − 4·26-s − 4·27-s − 8·29-s − 8·30-s + 8·31-s + 6·32-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 − 1.80·11-s − 1.73·12-s − 0.554·13-s − 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 0.917·19-s + 1.34·20-s − 2.55·22-s − 2.50·23-s − 1.63·24-s − 7/5·25-s − 0.784·26-s − 0.769·27-s − 1.48·29-s − 1.46·30-s + 1.43·31-s + 1.06·32-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: \(2\)
Selberg data: \((4,\ 24980004,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
11$D_{4}$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bd
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_t
19$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_y
23$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_cw
29$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cu
31$D_{4}$ \( 1 - 8 T + 60 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_ci
37$D_{4}$ \( 1 + 10 T + 67 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_cp
41$D_{4}$ \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_cq
43$D_{4}$ \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_ef
47$D_{4}$ \( 1 + 16 T + 140 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.47.q_fk
53$D_{4}$ \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.53.ac_dv
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.61.am_ew
67$D_{4}$ \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_cj
71$D_{4}$ \( 1 - 8 T - 4 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_ae
73$D_{4}$ \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.73.ao_gh
79$D_{4}$ \( 1 + 10 T + 181 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_gz
83$D_{4}$ \( 1 + 14 T + 213 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.83.o_if
89$D_{4}$ \( 1 + 22 T + 3 p T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.89.w_kh
97$D_{4}$ \( 1 + 6 T + 195 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_hn
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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

−7.900265414227790271720039308260, −7.60320776955292366652285919985, −7.22104362480009141757129153837, −6.86230406707792879823196882531, −6.34056837063142664947094121074, −6.15324973301255640771309066402, −5.63737658810082109413467162261, −5.55216263015654737539146589480, −5.20822995222430638991053522901, −5.02097481754984035721534323951, −4.42127110474716407397690003477, −4.07195053133499393012173342885, −3.58707330245954172800683386957, −3.21935439044866213925113963056, −2.54765220617574294118133168855, −2.29603981038630018112668121140, −1.62554399514333049626924143747, −1.51944590399535319527432851602, 0, 0, 1.51944590399535319527432851602, 1.62554399514333049626924143747, 2.29603981038630018112668121140, 2.54765220617574294118133168855, 3.21935439044866213925113963056, 3.58707330245954172800683386957, 4.07195053133499393012173342885, 4.42127110474716407397690003477, 5.02097481754984035721534323951, 5.20822995222430638991053522901, 5.55216263015654737539146589480, 5.63737658810082109413467162261, 6.15324973301255640771309066402, 6.34056837063142664947094121074, 6.86230406707792879823196882531, 7.22104362480009141757129153837, 7.60320776955292366652285919985, 7.900265414227790271720039308260

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