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Properties

Label 4-483e2-1.1-c1e2-0-3
Degree $4$
Conductor $233289$
Sign $1$
Analytic cond. $14.8747$
Root an. cond. $1.96386$
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 − 3-s + 2·4-s − 5-s − 6-s − 5·7-s + 5·8-s − 10-s + 2·11-s − 2·12-s + 14·13-s − 5·14-s + 15-s + 5·16-s + 3·17-s + 8·19-s − 2·20-s + 5·21-s + 2·22-s − 23-s − 5·24-s + 5·25-s + 14·26-s + 27-s − 10·28-s − 16·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.408·6-s − 1.88·7-s + 1.76·8-s − 0.316·10-s + 0.603·11-s − 0.577·12-s + 3.88·13-s − 1.33·14-s + 0.258·15-s + 5/4·16-s + 0.727·17-s + 1.83·19-s − 0.447·20-s + 1.09·21-s + 0.426·22-s − 0.208·23-s − 1.02·24-s + 25-s + 2.74·26-s + 0.192·27-s − 1.88·28-s − 2.97·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(233289\)    =    \(3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(14.8747\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 233289,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.751420178\)
\(L(\frac12)\) \(\approx\) \(2.751420178\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
23$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_ab
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.13.ao_cx
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.29.q_es
31$C_2^2$ \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_f
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_ab
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.41.i_du
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_acf
67$C_2^2$ \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) 2.67.b_aco
71$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.71.as_ip
73$C_2^2$ \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.73.d_acm
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.83.bk_sw
89$C_2^2$ \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.89.k_l
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−11.22609020206845286718559712019, −10.93505914489391503758988382557, −10.41595302224400769711577894662, −10.15823052177357403415408072689, −9.310185389742507261696250970402, −9.204408644377292953707691548368, −8.223722379947388569981491720133, −8.222214438262420353898893645951, −7.14150797216502735045310755629, −7.04303631641619256327903066812, −6.56704737547837718357305151790, −5.91653160692368706788214214778, −5.81947094035722219260171846294, −5.27185282501952250620263015009, −4.24077378684326536257538671452, −3.70534042441389695797843715171, −3.44541882281415437821136121111, −3.10504585523862376045047712238, −1.56704024946945987754513300783, −1.11229095968568495803319054532, 1.11229095968568495803319054532, 1.56704024946945987754513300783, 3.10504585523862376045047712238, 3.44541882281415437821136121111, 3.70534042441389695797843715171, 4.24077378684326536257538671452, 5.27185282501952250620263015009, 5.81947094035722219260171846294, 5.91653160692368706788214214778, 6.56704737547837718357305151790, 7.04303631641619256327903066812, 7.14150797216502735045310755629, 8.222214438262420353898893645951, 8.223722379947388569981491720133, 9.204408644377292953707691548368, 9.310185389742507261696250970402, 10.15823052177357403415408072689, 10.41595302224400769711577894662, 10.93505914489391503758988382557, 11.22609020206845286718559712019

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