Properties

Label 4-6864e2-1.1-c1e2-0-5
Degree $4$
Conductor $47114496$
Sign $1$
Analytic cond. $3004.06$
Root an. cond. $7.40333$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·5-s + 3·9-s + 2·11-s − 2·13-s + 8·15-s − 4·17-s + 4·23-s + 4·25-s − 4·27-s − 4·29-s + 8·31-s − 4·33-s − 4·37-s + 4·39-s − 4·41-s − 8·43-s − 12·45-s + 16·47-s − 6·49-s + 8·51-s − 4·53-s − 8·55-s − 12·61-s + 8·65-s − 8·69-s + 16·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 9-s + 0.603·11-s − 0.554·13-s + 2.06·15-s − 0.970·17-s + 0.834·23-s + 4/5·25-s − 0.769·27-s − 0.742·29-s + 1.43·31-s − 0.696·33-s − 0.657·37-s + 0.640·39-s − 0.624·41-s − 1.21·43-s − 1.78·45-s + 2.33·47-s − 6/7·49-s + 1.12·51-s − 0.549·53-s − 1.07·55-s − 1.53·61-s + 0.992·65-s − 0.963·69-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(47114496\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3004.06\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 47114496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_4$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 76 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 8 T + 84 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 108 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 286 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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

−7.74883021252135974222160138896, −7.47182294900589047155541576297, −6.90209548429201878135619780355, −6.83696367232453877081604308674, −6.47669262172993251323946996524, −6.21273687584574642043414012428, −5.48660584952677296108645475836, −5.33834051989332735929831736486, −4.90325343681183329991013133959, −4.52022094695823631568998793859, −4.12684405277911733872895397368, −4.06802266553851319487250904221, −3.31451382651844777120285372680, −3.28300473646686573795545199187, −2.45778113200640869140963331478, −2.03980195635201185504789478817, −1.33326412629552105195310475124, −0.869785714574265387661658123978, 0, 0, 0.869785714574265387661658123978, 1.33326412629552105195310475124, 2.03980195635201185504789478817, 2.45778113200640869140963331478, 3.28300473646686573795545199187, 3.31451382651844777120285372680, 4.06802266553851319487250904221, 4.12684405277911733872895397368, 4.52022094695823631568998793859, 4.90325343681183329991013133959, 5.33834051989332735929831736486, 5.48660584952677296108645475836, 6.21273687584574642043414012428, 6.47669262172993251323946996524, 6.83696367232453877081604308674, 6.90209548429201878135619780355, 7.47182294900589047155541576297, 7.74883021252135974222160138896

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