Properties

Label 4-3630e2-1.1-c1e2-0-14
Degree $4$
Conductor $13176900$
Sign $1$
Analytic cond. $840.170$
Root an. cond. $5.38383$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

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 − 6·12-s − 6·13-s − 4·15-s + 5·16-s − 12·17-s + 6·18-s − 6·19-s + 6·20-s + 6·23-s − 8·24-s + 3·25-s − 12·26-s − 4·27-s − 6·29-s − 8·30-s − 10·31-s + 6·32-s − 24·34-s + 9·36-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 − 1.73·12-s − 1.66·13-s − 1.03·15-s + 5/4·16-s − 2.91·17-s + 1.41·18-s − 1.37·19-s + 1.34·20-s + 1.25·23-s − 1.63·24-s + 3/5·25-s − 2.35·26-s − 0.769·27-s − 1.11·29-s − 1.46·30-s − 1.79·31-s + 1.06·32-s − 4.11·34-s + 3/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13176900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(840.170\)
Root analytic conductor: \(5.38383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13176900,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$D_{4}$ \( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.17.m_cp
19$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bj
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bc
29$D_{4}$ \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_cd
31$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_ci
37$D_{4}$ \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_bz
41$D_{4}$ \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_q
43$D_{4}$ \( 1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.43.g_do
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$D_{4}$ \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_dk
59$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_dw
61$D_{4}$ \( 1 + 6 T + 128 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_ey
67$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_bq
71$D_{4}$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_br
73$D_{4}$ \( 1 + 6 T + 152 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_fw
79$D_{4}$ \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.79.ay_le
83$D_{4}$ \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.83.s_jb
89$D_{4}$ \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_ec
97$D_{4}$ \( 1 - 10 T + 192 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_hk
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.126546771804876319163243814624, −8.014684757049097371758214995810, −7.05199973841545401532310316882, −7.03190620369022893967166947646, −6.80756788566478105013425310855, −6.53089740743057091084277917994, −6.02119886732138844052808609551, −5.73345137640759434287799783386, −5.10448638917216533959884876912, −4.98216074358931462677950766580, −4.70176258550413658560010032588, −4.46018989788031575541258359894, −3.60182253210008652045459675120, −3.56336897420761927912411027673, −2.57353098095275865165500781365, −2.35091975179907651328230673856, −1.78779427008832037712290472200, −1.62637966065797898038249460880, 0, 0, 1.62637966065797898038249460880, 1.78779427008832037712290472200, 2.35091975179907651328230673856, 2.57353098095275865165500781365, 3.56336897420761927912411027673, 3.60182253210008652045459675120, 4.46018989788031575541258359894, 4.70176258550413658560010032588, 4.98216074358931462677950766580, 5.10448638917216533959884876912, 5.73345137640759434287799783386, 6.02119886732138844052808609551, 6.53089740743057091084277917994, 6.80756788566478105013425310855, 7.03190620369022893967166947646, 7.05199973841545401532310316882, 8.014684757049097371758214995810, 8.126546771804876319163243814624

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