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Properties

Label 4-1512e2-1.1-c1e2-0-10
Degree $4$
Conductor $2286144$
Sign $1$
Analytic cond. $145.766$
Root an. cond. $3.47467$
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  − 4·5-s + 4·7-s + 2·11-s + 10·13-s + 6·17-s + 4·19-s + 6·23-s + 5·25-s − 12·29-s + 7·31-s − 16·35-s − 7·37-s − 4·41-s − 14·43-s − 2·47-s + 9·49-s + 6·53-s − 8·55-s + 6·59-s + 9·61-s − 40·65-s + 7·67-s + 16·71-s − 10·73-s + 8·77-s − 79-s − 28·83-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.51·7-s + 0.603·11-s + 2.77·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 25-s − 2.22·29-s + 1.25·31-s − 2.70·35-s − 1.15·37-s − 0.624·41-s − 2.13·43-s − 0.291·47-s + 9/7·49-s + 0.824·53-s − 1.07·55-s + 0.781·59-s + 1.15·61-s − 4.96·65-s + 0.855·67-s + 1.89·71-s − 1.17·73-s + 0.911·77-s − 0.112·79-s − 3.07·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(145.766\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.636946431\)
\(L(\frac12)\) \(\approx\) \(2.636946431\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_t
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_m
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.43.o_ff
47$C_2^2$ \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_abr
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ax
61$C_2^2$ \( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_u
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_as
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.k_bb
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) 2.79.b_ada
83$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.83.bc_ny
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_cd
97$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.97.be_qd
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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

−9.891103050997054822796870221026, −9.102578133149313291686741161861, −8.560629388506287812779049031959, −8.298330237285907210775313761393, −8.290977677570387414625359999282, −7.87074777864712119392797451388, −7.20213524718571884979162864814, −7.03371033028207807433694901342, −6.63604349944254916072148666766, −5.76266706380442238274247075175, −5.55159897711243042775569830405, −5.17272820491445669458447354842, −4.57937439209619076441370522601, −3.95378218285554347177572472636, −3.60755289181998327364524526290, −3.59989910700719067871166672192, −2.88706280955806689260401758043, −1.56298052508001374066927580267, −1.47932143978915576995787333997, −0.74239155082555296255416921322, 0.74239155082555296255416921322, 1.47932143978915576995787333997, 1.56298052508001374066927580267, 2.88706280955806689260401758043, 3.59989910700719067871166672192, 3.60755289181998327364524526290, 3.95378218285554347177572472636, 4.57937439209619076441370522601, 5.17272820491445669458447354842, 5.55159897711243042775569830405, 5.76266706380442238274247075175, 6.63604349944254916072148666766, 7.03371033028207807433694901342, 7.20213524718571884979162864814, 7.87074777864712119392797451388, 8.290977677570387414625359999282, 8.298330237285907210775313761393, 8.560629388506287812779049031959, 9.102578133149313291686741161861, 9.891103050997054822796870221026

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