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Properties

Label 4-147852-1.1-c1e2-0-10
Degree $4$
Conductor $147852$
Sign $1$
Analytic cond. $9.42717$
Root an. cond. $1.75224$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 6·7-s + 9-s − 12-s + 2·13-s + 16-s + 6·19-s − 6·21-s − 10·25-s − 27-s + 6·28-s − 12·31-s + 36-s − 2·37-s − 2·39-s + 24·43-s − 48-s + 13·49-s + 2·52-s − 6·57-s + 4·61-s + 6·63-s + 64-s + 4·67-s − 6·73-s + 10·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 2.26·7-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 1.37·19-s − 1.30·21-s − 2·25-s − 0.192·27-s + 1.13·28-s − 2.15·31-s + 1/6·36-s − 0.328·37-s − 0.320·39-s + 3.65·43-s − 0.144·48-s + 13/7·49-s + 0.277·52-s − 0.794·57-s + 0.512·61-s + 0.755·63-s + 1/8·64-s + 0.488·67-s − 0.702·73-s + 1.15·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147852 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147852 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(147852\)    =    \(2^{2} \cdot 3^{3} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(9.42717\)
Root analytic conductor: \(1.75224\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 147852,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.106698295\)
\(L(\frac12)\) \(\approx\) \(2.106698295\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 + T \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.a_v
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.19.ag_bv
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.53.a_eb
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.73.g_fz
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.a_gn
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−9.435475333424262584223799187666, −8.792199929808722838659257912612, −8.093256336428860279579050698932, −7.80630444762192273011758362778, −7.43313124434508075165402720464, −7.07433016066728121125499972487, −6.01497418650790910506005868954, −5.85671815365648243578194470848, −5.18717331204885286486593982487, −4.93990077281798148757107408401, −3.90575639700327756731471854422, −3.81933209527513554138512512650, −2.46404015546282379837947782123, −1.81998707337995438672490638604, −1.15779711357483022032503243329, 1.15779711357483022032503243329, 1.81998707337995438672490638604, 2.46404015546282379837947782123, 3.81933209527513554138512512650, 3.90575639700327756731471854422, 4.93990077281798148757107408401, 5.18717331204885286486593982487, 5.85671815365648243578194470848, 6.01497418650790910506005868954, 7.07433016066728121125499972487, 7.43313124434508075165402720464, 7.80630444762192273011758362778, 8.093256336428860279579050698932, 8.792199929808722838659257912612, 9.435475333424262584223799187666

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