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Properties

Label 4-165888-1.1-c1e2-0-11
Degree $4$
Conductor $165888$
Sign $1$
Analytic cond. $10.5771$
Root an. cond. $1.80340$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·11-s + 4·13-s + 2·25-s − 4·37-s + 16·47-s − 2·49-s + 16·59-s − 20·61-s − 16·71-s − 4·73-s − 8·83-s + 4·97-s − 16·107-s − 12·109-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  + 2.41·11-s + 1.10·13-s + 2/5·25-s − 0.657·37-s + 2.33·47-s − 2/7·49-s + 2.08·59-s − 2.56·61-s − 1.89·71-s − 0.468·73-s − 0.878·83-s + 0.406·97-s − 1.54·107-s − 1.14·109-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165888 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165888 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(165888\)    =    \(2^{11} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(10.5771\)
Root analytic conductor: \(1.80340\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 165888,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.085183990\)
\(L(\frac12)\) \(\approx\) \(2.085183990\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.11.ai_bi
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ae_o
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.29.a_o
31$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.31.a_abe
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.e_ck
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
43$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.43.a_co
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.53.a_be
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.59.aq_gw
61$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.u_hy
67$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.67.a_adq
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.q_hi
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.e_di
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.79.a_adq
83$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.i_de
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.89.a_de
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.126114682642108105410474176787, −8.816698349936449768745453830680, −8.490241311859140985938472760151, −7.74695028423085012602155920623, −7.14112329420953064038349678549, −6.80142119296241564499978441188, −6.24096722830303721273775957043, −5.90315234334121451845306922134, −5.28089485335483640646297616857, −4.32427486663986319186691113483, −4.10695357037614032838556007641, −3.53065418439902983766953623569, −2.80199447647835178093700059470, −1.67485090370693746162229758295, −1.10966450287732948909488876015, 1.10966450287732948909488876015, 1.67485090370693746162229758295, 2.80199447647835178093700059470, 3.53065418439902983766953623569, 4.10695357037614032838556007641, 4.32427486663986319186691113483, 5.28089485335483640646297616857, 5.90315234334121451845306922134, 6.24096722830303721273775957043, 6.80142119296241564499978441188, 7.14112329420953064038349678549, 7.74695028423085012602155920623, 8.490241311859140985938472760151, 8.816698349936449768745453830680, 9.126114682642108105410474176787

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