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Properties

Label 4-546e2-1.1-c1e2-0-19
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  + 2-s − 3-s − 6-s − 7-s − 8-s + 3·11-s + 5·13-s − 14-s − 16-s + 3·17-s − 5·19-s + 21-s + 3·22-s + 6·23-s + 24-s − 10·25-s + 5·26-s + 27-s + 9·29-s + 16·31-s − 3·33-s + 3·34-s − 8·37-s − 5·38-s − 5·39-s − 3·41-s + 42-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.904·11-s + 1.38·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s − 1.14·19-s + 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s − 2·25-s + 0.980·26-s + 0.192·27-s + 1.67·29-s + 2.87·31-s − 0.522·33-s + 0.514·34-s − 1.31·37-s − 0.811·38-s − 0.800·39-s − 0.468·41-s + 0.154·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.048584117\)
\(L(\frac12)\) \(\approx\) \(2.048584117\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.53.g_el
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.73.bg_pm
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.79.ba_mp
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.83.abk_sw
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2^2$ \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_gd
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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

−11.27467675444554506627348569976, −10.43665446237303050423048562125, −10.17921788330878460074369981724, −10.02605263889822091944870710653, −9.122340129898367408359917653256, −8.862927880834299839115338554495, −8.218760440740126732519333153441, −8.206461721416714192763360892477, −7.20688672234468133435500442623, −6.57202235152663544342250571000, −6.50643086392239607624689136621, −5.93282311916951321499209389756, −5.62708270514104307497978292971, −4.70971873196188448076294038605, −4.56427042930626092281714506410, −3.82168485186125513243235329820, −3.36605741552866436779474637925, −2.78155526401293814983481276927, −1.69814671503214991208220847581, −0.826149236098139701427696371742, 0.826149236098139701427696371742, 1.69814671503214991208220847581, 2.78155526401293814983481276927, 3.36605741552866436779474637925, 3.82168485186125513243235329820, 4.56427042930626092281714506410, 4.70971873196188448076294038605, 5.62708270514104307497978292971, 5.93282311916951321499209389756, 6.50643086392239607624689136621, 6.57202235152663544342250571000, 7.20688672234468133435500442623, 8.206461721416714192763360892477, 8.218760440740126732519333153441, 8.862927880834299839115338554495, 9.122340129898367408359917653256, 10.02605263889822091944870710653, 10.17921788330878460074369981724, 10.43665446237303050423048562125, 11.27467675444554506627348569976

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