Properties

Label 4-21e4-1.1-c3e2-0-8
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $677.032$
Root an. cond. $5.10096$
Motivic weight $3$
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  + 3·2-s + 5·4-s + 6·5-s + 27·8-s + 18·10-s + 6·11-s − 16·13-s + 69·16-s − 6·17-s − 64·19-s + 30·20-s + 18·22-s − 6·23-s − 166·25-s − 48·26-s + 252·29-s − 40·31-s + 27·32-s − 18·34-s − 248·37-s − 192·38-s + 162·40-s − 450·41-s + 376·43-s + 30·44-s − 18·46-s − 12·47-s + ⋯
L(s)  = 1  + 1.06·2-s + 5/8·4-s + 0.536·5-s + 1.19·8-s + 0.569·10-s + 0.164·11-s − 0.341·13-s + 1.07·16-s − 0.0856·17-s − 0.772·19-s + 0.335·20-s + 0.174·22-s − 0.0543·23-s − 1.32·25-s − 0.362·26-s + 1.61·29-s − 0.231·31-s + 0.149·32-s − 0.0907·34-s − 1.10·37-s − 0.819·38-s + 0.640·40-s − 1.71·41-s + 1.33·43-s + 0.102·44-s − 0.0576·46-s − 0.0372·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(677.032\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.641069399\)
\(L(\frac12)\) \(\approx\) \(5.641069399\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2$D_{4}$ \( 1 - 3 T + p^{2} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - 6 T + 202 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 6 T + 1246 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 16 T + 2406 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 9778 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 64 T + 6534 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 7870 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 252 T + 56446 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 40 T - 13890 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 248 T + 98214 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 450 T + 175642 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 376 T + 161526 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 12 T + 141790 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 1104 T + 602230 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 804 T + 380614 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 428 T + 425886 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 148 T + 440790 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 954 T + 13106 p T^{2} + 954 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1072 T + 1063278 T^{2} + 1072 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 572 T + 901662 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1944 T + 1957030 T^{2} - 1944 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 366 T + 1156090 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 808 T + 903054 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.90583753399711327422700441534, −10.29312676949778676866035382490, −10.24971021793301382508582500496, −9.902982592825101480440539840035, −8.900585609738174508222466442371, −8.790806704786590083336837896943, −8.199470125953334476160793993620, −7.46977265758075947592952363114, −7.21673092787523816716364967762, −6.62120739731328617301461187962, −6.19164934299687274337274066824, −5.53639256926580118373420190407, −5.20872362483547187682229648869, −4.65342779442294263022442699013, −3.98311374031340365598012413412, −3.80196629205402709802203622507, −2.80268322442694725187351413560, −2.16558080544106776519264121809, −1.68227702885123946317957318959, −0.63506450749297929860961787280, 0.63506450749297929860961787280, 1.68227702885123946317957318959, 2.16558080544106776519264121809, 2.80268322442694725187351413560, 3.80196629205402709802203622507, 3.98311374031340365598012413412, 4.65342779442294263022442699013, 5.20872362483547187682229648869, 5.53639256926580118373420190407, 6.19164934299687274337274066824, 6.62120739731328617301461187962, 7.21673092787523816716364967762, 7.46977265758075947592952363114, 8.199470125953334476160793993620, 8.790806704786590083336837896943, 8.900585609738174508222466442371, 9.902982592825101480440539840035, 10.24971021793301382508582500496, 10.29312676949778676866035382490, 10.90583753399711327422700441534

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