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Properties

Label 4-18252-1.1-c1e2-0-5
Degree $4$
Conductor $18252$
Sign $1$
Analytic cond. $1.16376$
Root an. cond. $1.03864$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 9-s + 12-s + 2·13-s + 16-s − 6·17-s + 2·25-s + 27-s + 6·29-s + 36-s + 2·39-s + 4·43-s + 48-s − 10·49-s − 6·51-s + 2·52-s − 18·53-s − 8·61-s + 64-s − 6·68-s + 2·75-s − 8·79-s + 81-s + 6·87-s + 2·100-s + 6·101-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 2/5·25-s + 0.192·27-s + 1.11·29-s + 1/6·36-s + 0.320·39-s + 0.609·43-s + 0.144·48-s − 1.42·49-s − 0.840·51-s + 0.277·52-s − 2.47·53-s − 1.02·61-s + 1/8·64-s − 0.727·68-s + 0.230·75-s − 0.900·79-s + 1/9·81-s + 0.643·87-s + 1/5·100-s + 0.597·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18252\)    =    \(2^{2} \cdot 3^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1.16376\)
Root analytic conductor: \(1.03864\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 18252,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.502228045\)
\(L(\frac12)\) \(\approx\) \(1.502228045\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.37.a_ao
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.47.a_bi
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.s_gw
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.83.a_cs
89$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.89.a_bu
97$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.97.a_bi
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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

−10.93563518258845713120845017688, −10.59475755535623012184334224791, −9.708369190831748135945548633922, −9.430907642722528638762170531180, −8.572458409973526554506755045658, −8.395183795438829852494335205743, −7.63479972642725055644481206037, −7.00718071155814271187890809702, −6.42114607026630267960665820988, −6.00324506053438724220127539666, −4.85577777121988293348093740487, −4.40939668586775540080383135804, −3.38701266821518968242899296207, −2.70209032313917213425334155866, −1.65931682821513182800029485810, 1.65931682821513182800029485810, 2.70209032313917213425334155866, 3.38701266821518968242899296207, 4.40939668586775540080383135804, 4.85577777121988293348093740487, 6.00324506053438724220127539666, 6.42114607026630267960665820988, 7.00718071155814271187890809702, 7.63479972642725055644481206037, 8.395183795438829852494335205743, 8.572458409973526554506755045658, 9.430907642722528638762170531180, 9.708369190831748135945548633922, 10.59475755535623012184334224791, 10.93563518258845713120845017688

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