Properties

Label 4-18e2-1.1-c6e2-0-0
Degree $4$
Conductor $324$
Sign $1$
Analytic cond. $17.1476$
Root an. cond. $2.03493$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·4-s − 968·7-s + 6.73e3·13-s + 1.02e3·16-s + 1.14e4·19-s + 992·25-s + 3.09e4·28-s − 7.95e4·31-s + 1.05e5·37-s + 7.60e3·43-s + 4.67e5·49-s − 2.15e5·52-s + 2.65e4·61-s − 3.27e4·64-s + 3.37e5·67-s + 4.72e5·73-s − 3.67e5·76-s − 7.02e4·79-s − 6.52e6·91-s − 6.42e5·97-s − 3.17e4·100-s + 3.98e6·103-s + 3.88e5·109-s − 9.91e5·112-s + 1.74e6·121-s + 2.54e6·124-s + 127-s + ⋯
L(s)  = 1  − 1/2·4-s − 2.82·7-s + 3.06·13-s + 1/4·16-s + 1.67·19-s + 0.0634·25-s + 1.41·28-s − 2.67·31-s + 2.07·37-s + 0.0955·43-s + 3.97·49-s − 1.53·52-s + 0.116·61-s − 1/8·64-s + 1.12·67-s + 1.21·73-s − 0.837·76-s − 0.142·79-s − 8.65·91-s − 0.704·97-s − 0.0317·100-s + 3.64·103-s + 0.300·109-s − 0.705·112-s + 0.985·121-s + 1.33·124-s − 4.72·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(17.1476\)
Root analytic conductor: \(2.03493\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{18} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.186316008\)
\(L(\frac12)\) \(\approx\) \(1.186316008\)
\(L(4)\) 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)$
bad2$C_2$ \( 1 + p^{5} T^{2} \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 992 T^{2} + p^{12} T^{4} \)
7$C_2$ \( ( 1 + 484 T + p^{6} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 1745714 T^{2} + p^{12} T^{4} \)
13$C_2$ \( ( 1 - 3368 T + p^{6} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 48274976 T^{2} + p^{12} T^{4} \)
19$C_2$ \( ( 1 - 5744 T + p^{6} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 284666690 T^{2} + p^{12} T^{4} \)
29$C_2^2$ \( 1 - 327940544 T^{2} + p^{12} T^{4} \)
31$C_2$ \( ( 1 + 39796 T + p^{6} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 52526 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8128061984 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 - 3800 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 15661450658 T^{2} + p^{12} T^{4} \)
53$C_2^2$ \( 1 + 12666935680 T^{2} + p^{12} T^{4} \)
59$C_2^2$ \( 1 - 21940729490 T^{2} + p^{12} T^{4} \)
61$C_2$ \( ( 1 - 13250 T + p^{6} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 168968 T + p^{6} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 26256724990 T^{2} + p^{12} T^{4} \)
73$C_2$ \( ( 1 - 236144 T + p^{6} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 35116 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 653760187346 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 977236742720 T^{2} + p^{12} T^{4} \)
97$C_2$ \( ( 1 + 321424 T + p^{6} T^{2} )^{2} \)
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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

−18.06910386729150301377131706949, −16.62029485204699025951331168978, −16.32436931584549506497081861955, −15.96786541734827632151247851394, −15.37463139302735411915003575388, −14.12812289850175296566168762207, −13.51344595741453586559357503664, −12.91917231506903117264513196133, −12.79943948634922932546375385271, −11.43224405226010532435699155407, −10.75716920754941246673460170820, −9.630622809509253975325652294077, −9.404076889061244728090908165569, −8.549264004115012358299792564715, −7.23424034812212747387496025669, −6.19948274389871141938995126915, −5.80918410576232217533082322387, −3.63132619809715011724446302242, −3.40488705269129499046290458171, −0.77636111964462645662059400391, 0.77636111964462645662059400391, 3.40488705269129499046290458171, 3.63132619809715011724446302242, 5.80918410576232217533082322387, 6.19948274389871141938995126915, 7.23424034812212747387496025669, 8.549264004115012358299792564715, 9.404076889061244728090908165569, 9.630622809509253975325652294077, 10.75716920754941246673460170820, 11.43224405226010532435699155407, 12.79943948634922932546375385271, 12.91917231506903117264513196133, 13.51344595741453586559357503664, 14.12812289850175296566168762207, 15.37463139302735411915003575388, 15.96786541734827632151247851394, 16.32436931584549506497081861955, 16.62029485204699025951331168978, 18.06910386729150301377131706949

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