Properties

Label 4-1216e2-1.1-c3e2-0-6
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $5147.53$
Root an. cond. $8.47032$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5·5-s + 6·7-s − 39·9-s − 15·11-s + 59·13-s − 5·15-s − 104·17-s − 38·19-s − 6·21-s + 21·23-s − 217·25-s + 52·27-s + 137·29-s − 4·31-s + 15·33-s + 30·35-s + 152·37-s − 59·39-s − 210·41-s + 67·43-s − 195·45-s − 273·47-s − 431·49-s + 104·51-s − 209·53-s − 75·55-s + ⋯
L(s)  = 1  − 0.192·3-s + 0.447·5-s + 0.323·7-s − 1.44·9-s − 0.411·11-s + 1.25·13-s − 0.0860·15-s − 1.48·17-s − 0.458·19-s − 0.0623·21-s + 0.190·23-s − 1.73·25-s + 0.370·27-s + 0.877·29-s − 0.0231·31-s + 0.0791·33-s + 0.144·35-s + 0.675·37-s − 0.242·39-s − 0.799·41-s + 0.237·43-s − 0.645·45-s − 0.847·47-s − 1.25·49-s + 0.285·51-s − 0.541·53-s − 0.183·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5147.53\)
Root analytic conductor: \(8.47032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 + p T )^{2} \)
good3$D_{4}$ \( 1 + T + 40 T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 - p T + 242 T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 6 T + 467 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 15 T + 2020 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 59 T + 4566 T^{2} - 59 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 104 T + 11105 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 21 T + 8926 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 137 T + 1804 p T^{2} - 137 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 48414 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 152 T + 95910 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 210 T + 33442 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 67 T + 152598 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 273 T + 197422 T^{2} + 273 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 209 T + 308546 T^{2} + 209 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 799 T + 513800 T^{2} - 799 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 149 T + 419484 T^{2} + 149 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 3 p T + 611270 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 792 T + 730138 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 246 T + 719291 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 254 T + 817014 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 374 T + 840590 T^{2} - 374 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 564 T + 942034 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 178 T - 159966 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.887497458767636168413871775978, −8.837288451997550384778094492909, −8.334349528544635124455707922369, −8.103161055469253879596759956899, −7.66244134883072423161989835727, −6.96115110044977149974241134101, −6.40285147943909509277634607177, −6.33353283813303207043393214044, −5.75752419631031203929610974460, −5.52723084244230813672143173887, −4.72067022483333522048769755595, −4.66523560019041847585610278691, −3.74342898057647677627976120129, −3.53781171579627802275757303231, −2.64395079291579370652894924229, −2.42519103754010673657380446192, −1.73539415815282408875612601094, −1.12551167233205896775023929478, 0, 0, 1.12551167233205896775023929478, 1.73539415815282408875612601094, 2.42519103754010673657380446192, 2.64395079291579370652894924229, 3.53781171579627802275757303231, 3.74342898057647677627976120129, 4.66523560019041847585610278691, 4.72067022483333522048769755595, 5.52723084244230813672143173887, 5.75752419631031203929610974460, 6.33353283813303207043393214044, 6.40285147943909509277634607177, 6.96115110044977149974241134101, 7.66244134883072423161989835727, 8.103161055469253879596759956899, 8.334349528544635124455707922369, 8.837288451997550384778094492909, 8.887497458767636168413871775978

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