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Properties

Label 4-483e2-1.1-c1e2-0-9
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 2·4-s − 3·5-s − 2·6-s + 2·7-s − 3·8-s + 3·9-s − 3·10-s + 4·12-s − 7·13-s + 2·14-s + 6·15-s + 16-s − 2·17-s + 3·18-s − 12·19-s + 6·20-s − 4·21-s + 2·23-s + 6·24-s − 2·25-s − 7·26-s − 4·27-s − 4·28-s + 8·29-s + 6·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 4-s − 1.34·5-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 9-s − 0.948·10-s + 1.15·12-s − 1.94·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 2.75·19-s + 1.34·20-s − 0.872·21-s + 0.417·23-s + 1.22·24-s − 2/5·25-s − 1.37·26-s − 0.769·27-s − 0.755·28-s + 1.48·29-s + 1.09·30-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: \(2\)
Selberg data: \((4,\ 233289,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
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$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_l
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.11.a_r
13$D_{4}$ \( 1 + 7 T + 27 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.13.h_bb
17$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_p
19$D_{4}$ \( 1 + 12 T + 69 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.19.m_cr
29$D_{4}$ \( 1 - 8 T + p T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_bd
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$D_{4}$ \( 1 + 10 T + 79 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.37.k_db
41$D_{4}$ \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_bp
43$D_{4}$ \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_z
47$D_{4}$ \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_du
53$D_{4}$ \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.53.f_eh
59$D_{4}$ \( 1 + 3 T + 109 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.59.d_ef
61$D_{4}$ \( 1 + 7 T + 103 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_dz
67$D_{4}$ \( 1 + 7 T + 135 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_ff
71$D_{4}$ \( 1 - 21 T + 251 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.71.av_jr
73$D_{4}$ \( 1 - 2 T + 127 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_ex
79$D_{4}$ \( 1 + 6 T + 87 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_dj
83$D_{4}$ \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.83.c_dj
89$D_{4}$ \( 1 + 17 T + 189 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.89.r_hh
97$D_{4}$ \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_eb
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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

−10.79450593174039892510984598321, −10.70132206775849479650400765813, −9.893281986431813321615983652435, −9.505786204436346415991662290088, −8.847186422737230578667744672458, −8.546331384402012229223328991291, −7.968331840466953306972192152846, −7.60114981634264924556613022056, −6.87542569106972419037511880281, −6.75100769493312226531274994752, −5.84992407040168717391639404792, −5.36091861991242819198887299311, −4.83836721377746536790923230274, −4.59403386190936932432158710194, −4.00687879834158679166073984332, −3.92484120595962203859369866348, −2.66890882043305631932503004946, −1.82813868592499159551825674143, 0, 0, 1.82813868592499159551825674143, 2.66890882043305631932503004946, 3.92484120595962203859369866348, 4.00687879834158679166073984332, 4.59403386190936932432158710194, 4.83836721377746536790923230274, 5.36091861991242819198887299311, 5.84992407040168717391639404792, 6.75100769493312226531274994752, 6.87542569106972419037511880281, 7.60114981634264924556613022056, 7.968331840466953306972192152846, 8.546331384402012229223328991291, 8.847186422737230578667744672458, 9.505786204436346415991662290088, 9.893281986431813321615983652435, 10.70132206775849479650400765813, 10.79450593174039892510984598321

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