Properties

Label 4-18e4-1.1-c1e2-0-1
Degree $4$
Conductor $104976$
Sign $1$
Analytic cond. $6.69336$
Root an. cond. $1.60846$
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  − 5·7-s + 7·13-s − 2·19-s + 5·25-s + 4·31-s − 2·37-s − 8·43-s + 7·49-s + 13·61-s − 11·67-s + 34·73-s + 13·79-s − 35·91-s − 5·97-s + 7·103-s + 4·109-s + 11·121-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.94·13-s − 0.458·19-s + 25-s + 0.718·31-s − 0.328·37-s − 1.21·43-s + 49-s + 1.66·61-s − 1.34·67-s + 3.97·73-s + 1.46·79-s − 3.66·91-s − 0.507·97-s + 0.689·103-s + 0.383·109-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(6.69336\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 104976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.238574621\)
\(L(\frac12)\) \(\approx\) \(1.238574621\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.f_s
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.ah_bk
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.an_ee
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.l_cc
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.73.abi_qt
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.f_acu
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

−11.92893567873738895808040613308, −11.12611800332127043043445309373, −11.03604763223208315750936001424, −10.46900672684705922151516453328, −9.800061902910841349221896659816, −9.734468935517001051997820790788, −9.021788410912916157624860181979, −8.499816978974123648859951075744, −8.342785903395719378047634157708, −7.50780876952033811786354442138, −6.64253610396501901993766338709, −6.61546418584568748996332354964, −6.20200305017867850384522648625, −5.52542362782316614216500455019, −4.84678644916868500454409584126, −4.01074531449814063409273287254, −3.40833744756611229016433241825, −3.17112831268645506320487135411, −2.10046681190686105816304517361, −0.825202739181669871320106696033, 0.825202739181669871320106696033, 2.10046681190686105816304517361, 3.17112831268645506320487135411, 3.40833744756611229016433241825, 4.01074531449814063409273287254, 4.84678644916868500454409584126, 5.52542362782316614216500455019, 6.20200305017867850384522648625, 6.61546418584568748996332354964, 6.64253610396501901993766338709, 7.50780876952033811786354442138, 8.342785903395719378047634157708, 8.499816978974123648859951075744, 9.021788410912916157624860181979, 9.734468935517001051997820790788, 9.800061902910841349221896659816, 10.46900672684705922151516453328, 11.03604763223208315750936001424, 11.12611800332127043043445309373, 11.92893567873738895808040613308

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