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Properties

Label 4-2940e2-1.1-c1e2-0-23
Degree $4$
Conductor $8643600$
Sign $1$
Analytic cond. $551.123$
Root an. cond. $4.84520$
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  − 2·3-s + 2·5-s + 3·9-s − 4·11-s − 4·15-s + 8·17-s − 8·19-s − 12·23-s + 3·25-s − 4·27-s − 4·29-s + 8·33-s − 4·37-s + 4·41-s − 8·43-s + 6·45-s + 4·47-s − 16·51-s − 12·53-s − 8·55-s + 16·57-s − 8·67-s + 24·69-s − 4·71-s + 8·73-s − 6·75-s − 20·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 9-s − 1.20·11-s − 1.03·15-s + 1.94·17-s − 1.83·19-s − 2.50·23-s + 3/5·25-s − 0.769·27-s − 0.742·29-s + 1.39·33-s − 0.657·37-s + 0.624·41-s − 1.21·43-s + 0.894·45-s + 0.583·47-s − 2.24·51-s − 1.64·53-s − 1.07·55-s + 2.11·57-s − 0.977·67-s + 2.88·69-s − 0.474·71-s + 0.936·73-s − 0.692·75-s − 2.25·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8643600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(551.123\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8643600,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bq
19$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_ca
23$D_{4}$ \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.23.m_dc
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_be
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_g
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cs
47$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_ba
53$D_{4}$ \( 1 + 12 T + 140 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_fk
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.59.a_eg
61$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.61.a_y
67$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_fm
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_cw
73$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_dm
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$D_{4}$ \( 1 - 4 T + 162 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_gg
89$D_{4}$ \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_gs
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.97.i_ic
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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

−8.434904706409245041803606085114, −8.089145571443959815122804193585, −7.77336159544666590937131451807, −7.60218206053875117319060066753, −6.76276359566886142169990391733, −6.72536297997960412923404937926, −5.98641290225549443638293617610, −5.98184181601309607716538431783, −5.49608852287800324602289219882, −5.36316998166682130477817503670, −4.64675656007202223505177728368, −4.47373044270309802727150957497, −3.69922434645829407315593768560, −3.57281874324961148022400139718, −2.52946727287980742198682798437, −2.46164090600680784712665182712, −1.50383694567677316079060801062, −1.47575244063771902260922442723, 0, 0, 1.47575244063771902260922442723, 1.50383694567677316079060801062, 2.46164090600680784712665182712, 2.52946727287980742198682798437, 3.57281874324961148022400139718, 3.69922434645829407315593768560, 4.47373044270309802727150957497, 4.64675656007202223505177728368, 5.36316998166682130477817503670, 5.49608852287800324602289219882, 5.98184181601309607716538431783, 5.98641290225549443638293617610, 6.72536297997960412923404937926, 6.76276359566886142169990391733, 7.60218206053875117319060066753, 7.77336159544666590937131451807, 8.089145571443959815122804193585, 8.434904706409245041803606085114

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