↑

Properties

Label 4-4788e2-1.1-c1e2-0-14
Degree $4$
Conductor $22924944$
Sign $1$
Analytic cond. $1461.71$
Root an. cond. $6.18323$
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  + 2·7-s − 4·11-s + 2·19-s − 4·23-s − 8·25-s − 4·29-s − 4·31-s + 4·37-s − 8·41-s + 4·43-s + 3·49-s − 4·53-s − 16·59-s − 4·61-s + 8·67-s − 28·71-s − 12·73-s − 8·77-s + 24·79-s + 8·89-s + 4·97-s − 8·101-s + 16·103-s − 28·107-s + 12·109-s − 12·113-s − 10·121-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.20·11-s + 0.458·19-s − 0.834·23-s − 8/5·25-s − 0.742·29-s − 0.718·31-s + 0.657·37-s − 1.24·41-s + 0.609·43-s + 3/7·49-s − 0.549·53-s − 2.08·59-s − 0.512·61-s + 0.977·67-s − 3.32·71-s − 1.40·73-s − 0.911·77-s + 2.70·79-s + 0.847·89-s + 0.406·97-s − 0.796·101-s + 1.57·103-s − 2.70·107-s + 1.14·109-s − 1.12·113-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 22924944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22924944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(22924944\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1461.71\)
Root analytic conductor: \(6.18323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 22924944,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_s
29$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_ci
31$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_cg
37$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_g
41$D_{4}$ \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_co
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_m
59$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_fu
61$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_eo
67$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_da
71$D_{4}$ \( 1 + 28 T + 336 T^{2} + 28 p T^{3} + p^{2} T^{4} \) 2.71.bc_my
73$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_fu
79$D_{4}$ \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.79.ay_li
83$C_2^2$ \( 1 + 164 T^{2} + p^{2} T^{4} \) 2.83.a_gi
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.89.ai_hm
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_gk
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

−7.87310362529097307416656725060, −7.74842576303359474357358729524, −7.47006750839650680017319183741, −7.38459854021968739465688102888, −6.44395940653967777455071421527, −6.36071377448973138260891710138, −5.78156473253580039371533791354, −5.64823959388904166318957747919, −5.07566147083065032175335697896, −4.88558181924899562483121931182, −4.42293299928465566223611421464, −3.92077519481118726494592225486, −3.59405683020532406293635415270, −3.14298853307600965889940771986, −2.38030094933027795975753773356, −2.37745428103109298094846144212, −1.52830066126480666744683537254, −1.36423693170917507869562531981, 0, 0, 1.36423693170917507869562531981, 1.52830066126480666744683537254, 2.37745428103109298094846144212, 2.38030094933027795975753773356, 3.14298853307600965889940771986, 3.59405683020532406293635415270, 3.92077519481118726494592225486, 4.42293299928465566223611421464, 4.88558181924899562483121931182, 5.07566147083065032175335697896, 5.64823959388904166318957747919, 5.78156473253580039371533791354, 6.36071377448973138260891710138, 6.44395940653967777455071421527, 7.38459854021968739465688102888, 7.47006750839650680017319183741, 7.74842576303359474357358729524, 7.87310362529097307416656725060

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