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Properties

Label 4-1596e2-1.1-c1e2-0-17
Degree $4$
Conductor $2547216$
Sign $1$
Analytic cond. $162.412$
Root an. cond. $3.56989$
Motivic weight $1$
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  − 2·3-s − 2·5-s − 2·7-s + 3·9-s + 4·15-s + 6·17-s + 2·19-s + 4·21-s − 4·27-s − 14·29-s − 12·31-s + 4·35-s + 4·41-s + 8·43-s − 6·45-s − 14·47-s + 3·49-s − 12·51-s − 22·53-s − 4·57-s − 16·59-s + 8·61-s − 6·63-s + 4·67-s − 2·71-s − 4·73-s + 4·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 1.03·15-s + 1.45·17-s + 0.458·19-s + 0.872·21-s − 0.769·27-s − 2.59·29-s − 2.15·31-s + 0.676·35-s + 0.624·41-s + 1.21·43-s − 0.894·45-s − 2.04·47-s + 3/7·49-s − 1.68·51-s − 3.02·53-s − 0.529·57-s − 2.08·59-s + 1.02·61-s − 0.755·63-s + 0.488·67-s − 0.237·71-s − 0.468·73-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2547216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2547216\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(162.412\)
Root analytic conductor: \(3.56989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2547216,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_e
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 - 6 T + 36 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bk
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$D_{4}$ \( 1 + 14 T + 100 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.29.o_dw
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.37.a_bu
41$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_cg
43$D_{4}$ \( 1 - 8 T + 74 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_cw
47$D_{4}$ \( 1 + 14 T + 136 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.47.o_fg
53$D_{4}$ \( 1 + 22 T + 220 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.53.w_im
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_eg
67$D_{4}$ \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_eg
71$D_{4}$ \( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_dc
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$D_{4}$ \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_fe
83$D_{4}$ \( 1 + 14 T + 152 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.83.o_fw
89$D_{4}$ \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_fy
97$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_eo
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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

−9.339660474161461120636220087082, −9.095404117052278066882636120250, −8.134121696462176191346111233310, −7.914694149073039160979278689362, −7.59969711716701668627208019443, −7.25242574115377694639789675226, −6.80335279120636741817316822910, −6.37614327417775029406458925310, −5.77952771226973842038975912271, −5.59819453573176297850932453059, −5.25325950857485524914760092113, −4.68355542982802265133587318492, −3.90649917702764626579501278689, −3.88402601791602431772753182247, −3.28064456219797238554127980994, −2.81252560660696671609277024490, −1.68902297744599191847556173748, −1.37175055495634852183239455558, 0, 0, 1.37175055495634852183239455558, 1.68902297744599191847556173748, 2.81252560660696671609277024490, 3.28064456219797238554127980994, 3.88402601791602431772753182247, 3.90649917702764626579501278689, 4.68355542982802265133587318492, 5.25325950857485524914760092113, 5.59819453573176297850932453059, 5.77952771226973842038975912271, 6.37614327417775029406458925310, 6.80335279120636741817316822910, 7.25242574115377694639789675226, 7.59969711716701668627208019443, 7.914694149073039160979278689362, 8.134121696462176191346111233310, 9.095404117052278066882636120250, 9.339660474161461120636220087082

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