Properties

Label 4-819e2-1.1-c1e2-0-12
Degree $4$
Conductor $670761$
Sign $1$
Analytic cond. $42.7683$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·13-s − 4·16-s + 4·17-s + 2·23-s + 25-s − 10·29-s + 18·43-s − 49-s − 18·53-s − 16·61-s + 30·79-s + 16·101-s − 32·103-s + 24·107-s + 42·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + ⋯
L(s)  = 1  + 1.10·13-s − 16-s + 0.970·17-s + 0.417·23-s + 1/5·25-s − 1.85·29-s + 2.74·43-s − 1/7·49-s − 2.47·53-s − 2.04·61-s + 3.37·79-s + 1.59·101-s − 3.15·103-s + 2.32·107-s + 3.95·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + 0.0760·173-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 670761,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.928740285\)
\(L(\frac12)\) \(\approx\) \(1.928740285\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
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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

−10.86886749961792428436370057998, −9.829263423240537081908856520972, −9.611194872241140155710546806198, −9.137505264505083486332439864372, −8.904367246599325213759210513815, −8.349483547880858567401818051772, −7.83092885676215183737260321766, −7.36806488411198938482497420710, −7.25526601724211495470022035448, −6.37652777732491049683915618624, −5.93630370935974896079491771885, −5.93215685579438675684257100549, −4.95002915943273554773508624904, −4.73694598141883719712751855191, −4.02936398096495684560903501212, −3.47734682414405739433088248348, −3.11879311128771217160959458667, −2.22725512495290528709750834833, −1.64514815958258871066860468702, −0.71299461444317999534864209824, 0.71299461444317999534864209824, 1.64514815958258871066860468702, 2.22725512495290528709750834833, 3.11879311128771217160959458667, 3.47734682414405739433088248348, 4.02936398096495684560903501212, 4.73694598141883719712751855191, 4.95002915943273554773508624904, 5.93215685579438675684257100549, 5.93630370935974896079491771885, 6.37652777732491049683915618624, 7.25526601724211495470022035448, 7.36806488411198938482497420710, 7.83092885676215183737260321766, 8.349483547880858567401818051772, 8.904367246599325213759210513815, 9.137505264505083486332439864372, 9.611194872241140155710546806198, 9.829263423240537081908856520972, 10.86886749961792428436370057998

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