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Properties

Label 4-2277e2-1.1-c1e2-0-16
Degree $4$
Conductor $5184729$
Sign $1$
Analytic cond. $330.582$
Root an. cond. $4.26402$
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  + 2-s − 2·4-s + 2·5-s + 4·7-s − 3·8-s + 2·10-s − 2·11-s + 12·13-s + 4·14-s + 16-s + 4·17-s + 4·19-s − 4·20-s − 2·22-s − 2·23-s − 7·25-s + 12·26-s − 8·28-s − 12·29-s + 2·32-s + 4·34-s + 8·35-s + 14·37-s + 4·38-s − 6·40-s + 6·41-s + 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s + 0.894·5-s + 1.51·7-s − 1.06·8-s + 0.632·10-s − 0.603·11-s + 3.32·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.917·19-s − 0.894·20-s − 0.426·22-s − 0.417·23-s − 7/5·25-s + 2.35·26-s − 1.51·28-s − 2.22·29-s + 0.353·32-s + 0.685·34-s + 1.35·35-s + 2.30·37-s + 0.648·38-s − 0.948·40-s + 0.937·41-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5184729\)    =    \(3^{4} \cdot 11^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(330.582\)
Root analytic conductor: \(4.26402\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5184729,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.157768215\)
\(L(\frac12)\) \(\approx\) \(5.157768215\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_n
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_s
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_w
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 + 57 T^{2} + p^{2} T^{4} \) 2.31.a_cf
37$D_{4}$ \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.37.ao_eo
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2^2$ \( 1 - 39 T^{2} + p^{2} T^{4} \) 2.43.a_abn
47$D_{4}$ \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_by
53$D_{4}$ \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_dj
59$D_{4}$ \( 1 + 2 T + 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_cw
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.67.abc_ms
71$D_{4}$ \( 1 - 14 T + 146 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.71.ao_fq
73$D_{4}$ \( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.73.o_hi
79$D_{4}$ \( 1 - 4 T + 157 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_gb
83$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_ha
89$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.89.ao_it
97$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_da
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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

−9.132691100390801487510301423218, −8.967878405385884590428071696254, −8.273812064959515633517042206060, −8.087245582166020587724991606495, −7.76754050236816710479025728100, −7.63012570156646872920410994359, −6.69055588989419331871445644213, −6.14738349962543185829017230632, −5.79713849414656788711930679215, −5.78704973532872455716945295972, −5.17546298305275656831704586431, −5.07257376629018513568883391594, −4.18154226755989617684829068055, −4.06884594574649080352202802084, −3.54469871346207516066358994208, −3.37666894307346969913258329260, −2.28962723072783962892520223976, −1.91855916077604607137348581883, −1.24488559625014395696513525311, −0.820476991937831851109502018246, 0.820476991937831851109502018246, 1.24488559625014395696513525311, 1.91855916077604607137348581883, 2.28962723072783962892520223976, 3.37666894307346969913258329260, 3.54469871346207516066358994208, 4.06884594574649080352202802084, 4.18154226755989617684829068055, 5.07257376629018513568883391594, 5.17546298305275656831704586431, 5.78704973532872455716945295972, 5.79713849414656788711930679215, 6.14738349962543185829017230632, 6.69055588989419331871445644213, 7.63012570156646872920410994359, 7.76754050236816710479025728100, 8.087245582166020587724991606495, 8.273812064959515633517042206060, 8.967878405385884590428071696254, 9.132691100390801487510301423218

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