Properties

Label 4-21e4-1.1-c3e2-0-12
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

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 8·4-s + 3·5-s − 45·8-s − 9·10-s − 15·11-s + 128·13-s + 135·16-s − 84·17-s − 16·19-s + 24·20-s + 45·22-s − 84·23-s + 125·25-s − 384·26-s + 594·29-s − 253·31-s − 360·32-s + 252·34-s + 316·37-s + 48·38-s − 135·40-s + 720·41-s + 52·43-s − 120·44-s + 252·46-s + 30·47-s + ⋯
L(s)  = 1  − 1.06·2-s + 4-s + 0.268·5-s − 1.98·8-s − 0.284·10-s − 0.411·11-s + 2.73·13-s + 2.10·16-s − 1.19·17-s − 0.193·19-s + 0.268·20-s + 0.436·22-s − 0.761·23-s + 25-s − 2.89·26-s + 3.80·29-s − 1.46·31-s − 1.98·32-s + 1.27·34-s + 1.40·37-s + 0.204·38-s − 0.533·40-s + 2.74·41-s + 0.184·43-s − 0.411·44-s + 0.807·46-s + 0.0931·47-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: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.107307497\)
\(L(\frac12)\) \(\approx\) \(2.107307497\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 3 T - 116 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 15 T - 1106 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 64 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 16 T - 6603 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 84 T - 5111 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 297 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 253 T + 34218 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 360 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 26 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T - 102923 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 363 T - 17108 T^{2} - 363 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 15 T - 205154 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 118 T - 213057 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 342 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 362 T - 257973 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 477 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 906 T + 115867 T^{2} + 906 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 503 T + p^{3} T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.88592742798339387609587095604, −10.60982644287911418758961908882, −9.973218517059621998057799642773, −9.519499412395209552185180453650, −8.910746689049461258871995135307, −8.777504961730791300813302332479, −8.302807898883038704078072954677, −8.077111420380656843208190387772, −7.22791433762193334569516139643, −6.62527404408692767441820137638, −6.22841474261099443111631796615, −6.08138782445709783735751988771, −5.49575018011669787358588336015, −4.52550649009489103195520527079, −3.98766008989132867855067248652, −3.23648330000092306167548948425, −2.67373217283019854892413570213, −2.12193959654287018804306110627, −0.920477360918718739067526671656, −0.77496663956091385349843618612, 0.77496663956091385349843618612, 0.920477360918718739067526671656, 2.12193959654287018804306110627, 2.67373217283019854892413570213, 3.23648330000092306167548948425, 3.98766008989132867855067248652, 4.52550649009489103195520527079, 5.49575018011669787358588336015, 6.08138782445709783735751988771, 6.22841474261099443111631796615, 6.62527404408692767441820137638, 7.22791433762193334569516139643, 8.077111420380656843208190387772, 8.302807898883038704078072954677, 8.777504961730791300813302332479, 8.910746689049461258871995135307, 9.519499412395209552185180453650, 9.973218517059621998057799642773, 10.60982644287911418758961908882, 10.88592742798339387609587095604

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