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Properties

Label 4-819e2-1.1-c1e2-0-27
Degree $4$
Conductor $670761$
Sign $1$
Analytic cond. $42.7683$
Root an. cond. $2.55729$
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  + 3·4-s + 4·7-s + 5·16-s − 6·25-s + 12·28-s + 4·37-s + 24·43-s + 9·49-s + 3·64-s − 16·67-s + 16·79-s − 18·100-s + 4·109-s + 20·112-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 12·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 72·172-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.51·7-s + 5/4·16-s − 6/5·25-s + 2.26·28-s + 0.657·37-s + 3.65·43-s + 9/7·49-s + 3/8·64-s − 1.95·67-s + 1.80·79-s − 9/5·100-s + 0.383·109-s + 1.88·112-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.986·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 5.48·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(670761\)    =    \(3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(42.7683\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 670761,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.810476638\)
\(L(\frac12)\) \(\approx\) \(3.810476638\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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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

−8.107530367726792542516902381016, −7.71581532580987418978840899910, −7.56326489158108072490065360833, −7.20120785357851515902900527767, −6.44593537941038145358398257586, −6.13949758592290680880334427678, −5.66670525826289776323190550935, −5.26964924686290152055608689787, −4.48371137978466660810322329493, −4.20847924863469053518436666093, −3.50377117937361779686405002252, −2.65229866390281895850843487513, −2.32898089052969175523726760080, −1.72566933363098237773624433718, −1.02882791170821988124391670848, 1.02882791170821988124391670848, 1.72566933363098237773624433718, 2.32898089052969175523726760080, 2.65229866390281895850843487513, 3.50377117937361779686405002252, 4.20847924863469053518436666093, 4.48371137978466660810322329493, 5.26964924686290152055608689787, 5.66670525826289776323190550935, 6.13949758592290680880334427678, 6.44593537941038145358398257586, 7.20120785357851515902900527767, 7.56326489158108072490065360833, 7.71581532580987418978840899910, 8.107530367726792542516902381016

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