Properties

Label 4-207e2-1.1-c5e2-0-0
Degree $4$
Conductor $42849$
Sign $1$
Analytic cond. $1102.20$
Root an. cond. $5.76189$
Motivic weight $5$
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  + 52·4-s − 94·5-s − 118·7-s − 320·11-s − 288·13-s + 1.68e3·16-s + 1.81e3·17-s + 730·19-s − 4.88e3·20-s − 1.05e3·23-s + 406·25-s − 6.13e3·28-s − 8.20e3·29-s + 1.77e3·31-s + 1.10e4·35-s − 2.31e4·37-s − 5.51e3·41-s + 1.03e4·43-s − 1.66e4·44-s − 4.29e4·47-s − 1.96e4·49-s − 1.49e4·52-s + 2.53e4·53-s + 3.00e4·55-s − 1.83e4·59-s + 3.72e4·61-s + 3.41e4·64-s + ⋯
L(s)  = 1  + 13/8·4-s − 1.68·5-s − 0.910·7-s − 0.797·11-s − 0.472·13-s + 1.64·16-s + 1.51·17-s + 0.463·19-s − 2.73·20-s − 0.417·23-s + 0.129·25-s − 1.47·28-s − 1.81·29-s + 0.331·31-s + 1.53·35-s − 2.77·37-s − 0.512·41-s + 0.851·43-s − 1.29·44-s − 2.83·47-s − 1.16·49-s − 0.768·52-s + 1.23·53-s + 1.34·55-s − 0.686·59-s + 1.28·61-s + 1.04·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42849 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(42849\)    =    \(3^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1102.20\)
Root analytic conductor: \(5.76189\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 42849,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.03489732462\)
\(L(\frac12)\) \(\approx\) \(0.03489732462\)
\(L(\frac{7}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 + p^{2} T )^{2} \)
good2$C_2^2$ \( 1 - 13 p^{2} T^{2} + p^{10} T^{4} \)
5$D_{4}$ \( 1 + 94 T + 1686 p T^{2} + 94 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 118 T + 4798 p T^{2} + 118 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 320 T + 9446 T^{2} + 320 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 288 T + 604518 T^{2} + 288 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 1810 T + 3653838 T^{2} - 1810 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 730 T + 2280514 T^{2} - 730 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 8208 T + 55813190 T^{2} + 8208 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 1772 T - 8430386 T^{2} - 1772 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 23112 T + 270998406 T^{2} + 23112 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 5516 T + 184729830 T^{2} + 5516 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 10322 T + 114379026 T^{2} - 10322 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 42952 T + 881422574 T^{2} + 42952 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 25350 T + 443489942 T^{2} - 25350 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 18344 T + 1506490326 T^{2} + 18344 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 37224 T + 1584042582 T^{2} - 37224 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 7482 T + 31324422 p T^{2} + 7482 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 126848 T + 7504629902 T^{2} + 126848 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 137660 T + 8741816710 T^{2} - 137660 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 62286 T + 7060563458 T^{2} - 62286 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 83120 T + 6016756982 T^{2} + 83120 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 69770 T + 12229395942 T^{2} + 69770 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 170104 T + 21347426014 T^{2} + 170104 p^{5} T^{3} + p^{10} T^{4} \)
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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

−11.83136901596434430941470193561, −11.19353850887812353827620284214, −11.11661753126810175058547655249, −10.30890215331346726658602466134, −9.848281256105182155587202461310, −9.628869262074978351165963348768, −8.449988699437276481110163418858, −8.082143968217803472809018330509, −7.66546723379361824426090925666, −7.07184315803980188190837272732, −7.04103836137771517446403231767, −6.10016702835673687079083655215, −5.56963573750612252820774097463, −5.00795625446597941440889590815, −3.88453505317408921205482118847, −3.34175584106067459612812913549, −3.18551410218446037596790780112, −2.15445826349197502502619695380, −1.43984990546038260019325864499, −0.05334564311679959179635229409, 0.05334564311679959179635229409, 1.43984990546038260019325864499, 2.15445826349197502502619695380, 3.18551410218446037596790780112, 3.34175584106067459612812913549, 3.88453505317408921205482118847, 5.00795625446597941440889590815, 5.56963573750612252820774097463, 6.10016702835673687079083655215, 7.04103836137771517446403231767, 7.07184315803980188190837272732, 7.66546723379361824426090925666, 8.082143968217803472809018330509, 8.449988699437276481110163418858, 9.628869262074978351165963348768, 9.848281256105182155587202461310, 10.30890215331346726658602466134, 11.11661753126810175058547655249, 11.19353850887812353827620284214, 11.83136901596434430941470193561

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