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Properties

Label 4-546e2-1.1-c1e2-0-44
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  + 3·3-s + 4-s + 2·7-s + 6·9-s + 3·12-s − 2·13-s + 16-s + 4·19-s + 6·21-s − 10·25-s + 9·27-s + 2·28-s + 2·31-s + 6·36-s + 14·37-s − 6·39-s − 24·43-s + 3·48-s + 3·49-s − 2·52-s + 12·57-s + 26·61-s + 12·63-s + 64-s + 22·67-s + 14·73-s − 30·75-s + ⋯
L(s)  = 1  + 1.73·3-s + 1/2·4-s + 0.755·7-s + 2·9-s + 0.866·12-s − 0.554·13-s + 1/4·16-s + 0.917·19-s + 1.30·21-s − 2·25-s + 1.73·27-s + 0.377·28-s + 0.359·31-s + 36-s + 2.30·37-s − 0.960·39-s − 3.65·43-s + 0.433·48-s + 3/7·49-s − 0.277·52-s + 1.58·57-s + 3.32·61-s + 1.51·63-s + 1/8·64-s + 2.68·67-s + 1.63·73-s − 3.46·75-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

\(L(1)\) \(\approx\) \(4.309745717\)
\(L(\frac12)\) \(\approx\) \(4.309745717\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.23.a_v
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.31.ac_cl
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.a_b
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.a_bt
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.61.aba_lf
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2$ \( ( 1 + 17 T + p T^{2} )^{2} \) 2.79.bi_rf
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−8.469152938024698877370110317561, −8.410332621243878174631771496694, −7.959201586617154759738546904094, −7.68409626813723475327909628947, −6.90643848137944117202320139495, −6.90516587374622685438056526369, −5.98244664513634320441588551536, −5.36737041037488579734906212522, −4.88738837750963504497893956492, −4.15982302022081941279948742151, −3.71179494687009087455073957292, −3.16464702872454555446785806554, −2.36943591207239051047957475876, −2.10681498002416439479097053291, −1.23766611312278004155286222469, 1.23766611312278004155286222469, 2.10681498002416439479097053291, 2.36943591207239051047957475876, 3.16464702872454555446785806554, 3.71179494687009087455073957292, 4.15982302022081941279948742151, 4.88738837750963504497893956492, 5.36737041037488579734906212522, 5.98244664513634320441588551536, 6.90516587374622685438056526369, 6.90643848137944117202320139495, 7.68409626813723475327909628947, 7.959201586617154759738546904094, 8.410332621243878174631771496694, 8.469152938024698877370110317561

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