↑

Properties

Label 4-4050e2-1.1-c1e2-0-24
Degree $4$
Conductor $16402500$
Sign $1$
Analytic cond. $1045.83$
Root an. cond. $5.68677$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 7-s + 4·8-s − 3·11-s + 2·13-s − 2·14-s + 5·16-s + 9·17-s + 19-s − 6·22-s + 3·23-s + 4·26-s − 3·28-s − 3·29-s − 2·31-s + 6·32-s + 18·34-s + 8·37-s + 2·38-s − 6·41-s + 17·43-s − 9·44-s + 6·46-s + 9·47-s − 5·49-s + 6·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s − 0.904·11-s + 0.554·13-s − 0.534·14-s + 5/4·16-s + 2.18·17-s + 0.229·19-s − 1.27·22-s + 0.625·23-s + 0.784·26-s − 0.566·28-s − 0.557·29-s − 0.359·31-s + 1.06·32-s + 3.08·34-s + 1.31·37-s + 0.324·38-s − 0.937·41-s + 2.59·43-s − 1.35·44-s + 0.884·46-s + 1.31·47-s − 5/7·49-s + 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16402500\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1045.83\)
Root analytic conductor: \(5.68677\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16402500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.245777164\)
\(L(\frac12)\) \(\approx\) \(9.245777164\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_g
11$D_{4}$ \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_q
13$D_{4}$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_ag
17$D_{4}$ \( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.17.aj_bu
19$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) 2.19.ab_be
23$D_{4}$ \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_bo
29$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ca
31$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_be
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$D_{4}$ \( 1 - 17 T + 150 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.43.ar_fu
47$D_{4}$ \( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_ec
53$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.53.a_aba
59$D_{4}$ \( 1 - 3 T + 112 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_ei
61$D_{4}$ \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) 2.61.ab_bw
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.67.ao_hb
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$D_{4}$ \( 1 - 11 T + 102 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_dy
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.79.ae_gg
83$D_{4}$ \( 1 + 9 T + 178 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_gw
89$D_{4}$ \( 1 - 15 T + 160 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_ge
97$D_{4}$ \( 1 - 11 T + 216 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.97.al_ii
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

−8.393654551240342517120479518039, −8.184879944692965017386934824812, −7.63630938807426205208881495949, −7.49889052682483060259552057809, −7.19140856055246670085717058036, −6.77219343662329818992683449477, −6.01520250820010483849625892974, −6.00614201168908602688651877628, −5.73264286940115781230820259714, −5.27099331596170313365126013316, −4.87916084681700607573524320408, −4.58719207551464155698526797573, −3.83952019781879064041204139785, −3.75198076459123684632222357284, −3.10430723010243721441050799738, −3.08055887254794659050315778346, −2.28268008657432106710816115058, −2.06195068169821969185207886849, −1.05166953128054851455476295020, −0.805764038373834662278906042714, 0.805764038373834662278906042714, 1.05166953128054851455476295020, 2.06195068169821969185207886849, 2.28268008657432106710816115058, 3.08055887254794659050315778346, 3.10430723010243721441050799738, 3.75198076459123684632222357284, 3.83952019781879064041204139785, 4.58719207551464155698526797573, 4.87916084681700607573524320408, 5.27099331596170313365126013316, 5.73264286940115781230820259714, 6.00614201168908602688651877628, 6.01520250820010483849625892974, 6.77219343662329818992683449477, 7.19140856055246670085717058036, 7.49889052682483060259552057809, 7.63630938807426205208881495949, 8.184879944692965017386934824812, 8.393654551240342517120479518039

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