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Properties

Label 4-84e2-1.1-c1e2-0-3
Degree $4$
Conductor $7056$
Sign $1$
Analytic cond. $0.449896$
Root an. cond. $0.818989$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 7-s − 2·11-s − 6·13-s + 2·15-s − 8·17-s + 19-s + 21-s − 8·23-s + 5·25-s − 27-s + 8·29-s − 3·31-s − 2·33-s + 2·35-s + 37-s − 6·39-s + 12·41-s + 22·43-s − 6·47-s − 6·49-s − 8·51-s + 12·53-s − 4·55-s + 57-s − 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s − 0.603·11-s − 1.66·13-s + 0.516·15-s − 1.94·17-s + 0.229·19-s + 0.218·21-s − 1.66·23-s + 25-s − 0.192·27-s + 1.48·29-s − 0.538·31-s − 0.348·33-s + 0.338·35-s + 0.164·37-s − 0.960·39-s + 1.87·41-s + 3.35·43-s − 0.875·47-s − 6/7·49-s − 1.12·51-s + 1.64·53-s − 0.539·55-s + 0.132·57-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.449896\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.129507333\)
\(L(\frac12)\) \(\approx\) \(1.129507333\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
17$C_2^2$ \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bv
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_bp
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_aw
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.ab_abk
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.43.aw_hz
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_al
53$C_2^2$ \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dn
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_az
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.67.n_dy
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.71.u_ji
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.73.al_bw
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.79.ad_acs
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.83.ae_go
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_li
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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

−14.34877846489454647415739956355, −14.25852321186986371167230880401, −13.36824278900699912279613894739, −13.23173806785161329894210906434, −12.29757996768351210651907455842, −12.16412271182825664677552101971, −11.14486027425762165114939820339, −10.69561359373933840087034095162, −10.12769028832827674560627736585, −9.491877347068163021800344931572, −9.081300569046940718317926347947, −8.435133764296601594801084367096, −7.66253600109065687526637660338, −7.24475308339824837646988399442, −6.30079322587263930132470688455, −5.71540711166325100898174515524, −4.74540548980467941915097919083, −4.27368775553655232306587359177, −2.59165756938551738616972537442, −2.31090490712168537397656940994, 2.31090490712168537397656940994, 2.59165756938551738616972537442, 4.27368775553655232306587359177, 4.74540548980467941915097919083, 5.71540711166325100898174515524, 6.30079322587263930132470688455, 7.24475308339824837646988399442, 7.66253600109065687526637660338, 8.435133764296601594801084367096, 9.081300569046940718317926347947, 9.491877347068163021800344931572, 10.12769028832827674560627736585, 10.69561359373933840087034095162, 11.14486027425762165114939820339, 12.16412271182825664677552101971, 12.29757996768351210651907455842, 13.23173806785161329894210906434, 13.36824278900699912279613894739, 14.25852321186986371167230880401, 14.34877846489454647415739956355

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