↑

Properties

Label 4-630e2-1.1-c1e2-0-63
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
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 + 3·3-s + 3·4-s + 5-s + 6·6-s − 4·7-s + 4·8-s + 6·9-s + 2·10-s + 9·12-s + 4·13-s − 8·14-s + 3·15-s + 5·16-s + 12·18-s − 2·19-s + 3·20-s − 12·21-s + 3·23-s + 12·24-s + 8·26-s + 9·27-s − 12·28-s − 6·29-s + 6·30-s + 4·31-s + 6·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.447·5-s + 2.44·6-s − 1.51·7-s + 1.41·8-s + 2·9-s + 0.632·10-s + 2.59·12-s + 1.10·13-s − 2.13·14-s + 0.774·15-s + 5/4·16-s + 2.82·18-s − 0.458·19-s + 0.670·20-s − 2.61·21-s + 0.625·23-s + 2.44·24-s + 1.56·26-s + 1.73·27-s − 2.26·28-s − 1.11·29-s + 1.09·30-s + 0.718·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.571061345\)
\(L(\frac12)\) \(\approx\) \(8.571061345\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_ao
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_abh
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.43.l_da
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ar
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.67.o_hb
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_acr
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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

−10.55356176675862854782324236799, −10.46190338462439932277847286279, −9.975280292588606817397177220846, −9.445033729965284520195419349202, −9.024300868298668252337431464618, −8.791522890318179597822811015662, −8.140950325875924112646368212599, −7.67287270462553408894963336702, −7.14190411991154944198345905385, −6.77327113186497511185373468528, −6.19246350730544000035282143921, −6.03093633686515806676927030567, −5.26629747983766552686987831856, −4.66090084177411322117201729768, −3.86387976294663709543627960013, −3.77702980781685845132090211119, −3.15936419588161073453390506441, −2.74891981447445444763778048110, −2.15580621080420397833725850882, −1.36481776270053802479604340386, 1.36481776270053802479604340386, 2.15580621080420397833725850882, 2.74891981447445444763778048110, 3.15936419588161073453390506441, 3.77702980781685845132090211119, 3.86387976294663709543627960013, 4.66090084177411322117201729768, 5.26629747983766552686987831856, 6.03093633686515806676927030567, 6.19246350730544000035282143921, 6.77327113186497511185373468528, 7.14190411991154944198345905385, 7.67287270462553408894963336702, 8.140950325875924112646368212599, 8.791522890318179597822811015662, 9.024300868298668252337431464618, 9.445033729965284520195419349202, 9.975280292588606817397177220846, 10.46190338462439932277847286279, 10.55356176675862854782324236799

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