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Properties

Label 4-18252-1.1-c1e2-0-0
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 + 12·23-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 − 6·53-s + 8·61-s − 64-s − 6·68-s − 12·69-s + 2·75-s + 8·79-s + 81-s − 6·87-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 + 2.50·23-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 − 0.824·53-s + 1.02·61-s − 1/8·64-s − 0.727·68-s − 1.44·69-s + 0.230·75-s + 0.900·79-s + 1/9·81-s − 0.643·87-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

\(L(1)\) \(\approx\) \(0.8686535141\)
\(L(\frac12)\) \(\approx\) \(0.8686535141\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
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_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
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$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
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_abi
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.g_bi
59$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.59.a_cw
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.ai_be
83$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.83.a_acs
89$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.89.a_abu
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

−10.97128464827036168503456499813, −10.38380426631300274507238830297, −9.910898509955468796291780417829, −9.375185120874788064000410662502, −8.922113478173866576244812971621, −8.073691663591242810368442583085, −7.70882818714125421774283383354, −6.92615545426317886994846021910, −6.47042859335959762859098560932, −5.54633011732372225630990817083, −5.10950048459928302879845079780, −4.57050866854444857687368672533, −3.55773110230824778345306246409, −2.79322709105415504521665964176, −1.14558033398518653925053000555, 1.14558033398518653925053000555, 2.79322709105415504521665964176, 3.55773110230824778345306246409, 4.57050866854444857687368672533, 5.10950048459928302879845079780, 5.54633011732372225630990817083, 6.47042859335959762859098560932, 6.92615545426317886994846021910, 7.70882818714125421774283383354, 8.073691663591242810368442583085, 8.922113478173866576244812971621, 9.375185120874788064000410662502, 9.910898509955468796291780417829, 10.38380426631300274507238830297, 10.97128464827036168503456499813

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