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Properties

Label 4-1725e2-1.1-c1e2-0-21
Degree $4$
Conductor $2975625$
Sign $1$
Analytic cond. $189.728$
Root an. cond. $3.71136$
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 − 2·4-s + 2·7-s + 3·9-s − 8·11-s − 4·12-s − 4·13-s − 2·17-s − 4·19-s + 4·21-s + 2·23-s + 4·27-s − 4·28-s − 10·29-s − 14·31-s − 16·33-s − 6·36-s − 6·37-s − 8·39-s − 18·41-s − 12·43-s + 16·44-s + 20·47-s − 3·49-s − 4·51-s + 8·52-s − 6·53-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 0.755·7-s + 9-s − 2.41·11-s − 1.15·12-s − 1.10·13-s − 0.485·17-s − 0.917·19-s + 0.872·21-s + 0.417·23-s + 0.769·27-s − 0.755·28-s − 1.85·29-s − 2.51·31-s − 2.78·33-s − 36-s − 0.986·37-s − 1.28·39-s − 2.81·41-s − 1.82·43-s + 2.41·44-s + 2.91·47-s − 3/7·49-s − 0.560·51-s + 1.10·52-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2975625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2975625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2975625\)    =    \(3^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(189.728\)
Root analytic conductor: \(3.71136\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2975625,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
7$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_h
11$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.11.i_bk
13$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_bc
17$D_{4}$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_ap
19$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_y
29$D_{4}$ \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_dd
31$D_{4}$ \( 1 + 14 T + 103 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.31.o_dz
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$D_{4}$ \( 1 + 18 T + 161 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.41.s_gf
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.43.m_es
47$D_{4}$ \( 1 - 20 T + 192 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.47.au_hk
53$D_{4}$ \( 1 + 6 T + 97 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_dt
59$D_{4}$ \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_bd
61$D_{4}$ \( 1 - 8 T + 136 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_fg
67$D_{4}$ \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_ct
71$D_{4}$ \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_bt
73$D_{4}$ \( 1 - 12 T + 164 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_gi
79$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_bu
83$D_{4}$ \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.83.o_gj
89$D_{4}$ \( 1 + 8 T - 6 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.89.i_ag
97$D_{4}$ \( 1 - 8 T + 202 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_hu
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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

−8.916918028928612182005946049004, −8.665138946810729755189058956118, −8.440044341229474346276289930110, −8.150379883685608462634338921737, −7.44656605637553718094564083988, −7.25787082802819912520353331785, −7.19613985301677964012109823597, −6.39538505541761392491487348212, −5.43929502302073579110953200378, −5.39765946907930750581724106742, −4.87959848517060035731484401520, −4.83103720978379443368330709331, −3.92056572605140958354773963068, −3.72039850382617311422326187905, −3.14729512317652580446090893469, −2.37537801167682862457734294859, −2.14245155958705174456577634757, −1.70530524565437787637489128732, 0, 0, 1.70530524565437787637489128732, 2.14245155958705174456577634757, 2.37537801167682862457734294859, 3.14729512317652580446090893469, 3.72039850382617311422326187905, 3.92056572605140958354773963068, 4.83103720978379443368330709331, 4.87959848517060035731484401520, 5.39765946907930750581724106742, 5.43929502302073579110953200378, 6.39538505541761392491487348212, 7.19613985301677964012109823597, 7.25787082802819912520353331785, 7.44656605637553718094564083988, 8.150379883685608462634338921737, 8.440044341229474346276289930110, 8.665138946810729755189058956118, 8.916918028928612182005946049004

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