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Properties

Label 4-60e3-1.1-c1e2-0-0
Degree $4$
Conductor $216000$
Sign $1$
Analytic cond. $13.7723$
Root an. cond. $1.92642$
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  − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s − 2·9-s + 10-s + 12-s + 15-s − 16-s + 2·18-s + 20-s − 3·24-s + 25-s + 5·27-s − 30-s + 7·31-s − 5·32-s + 2·36-s − 9·37-s − 3·40-s + 3·41-s − 6·43-s + 2·45-s + 48-s + 11·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s + 0.258·15-s − 1/4·16-s + 0.471·18-s + 0.223·20-s − 0.612·24-s + 1/5·25-s + 0.962·27-s − 0.182·30-s + 1.25·31-s − 0.883·32-s + 1/3·36-s − 1.47·37-s − 0.474·40-s + 0.468·41-s − 0.914·43-s + 0.298·45-s + 0.144·48-s + 11/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(216000\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(13.7723\)
Root analytic conductor: \(1.92642\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 216000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4745373084\)
\(L(\frac12)\) \(\approx\) \(0.4745373084\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3$C_2$ \( 1 + T + p T^{2} \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.17.a_ah
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.23.a_m
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.31.ah_cu
37$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.j_de
41$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ad_cu
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.g_bu
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.47.a_g
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.53.af_ce
59$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.59.a_p
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.67.y_kj
71$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.p_ew
73$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.73.a_afk
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.79.ao_hq
83$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.ap_im
89$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.p_hk
97$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.97.a_q
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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

−9.135302092333353652652186528333, −8.568822291342378708563372922224, −8.194417309724579054838462347537, −7.77531817530388024228130269937, −7.11431484901848047337285972916, −6.84899310638551761424185074277, −6.01612086119737147435067066575, −5.69886437390822823949011200716, −4.99207371310814885896269263171, −4.63021633725152443209733084264, −4.01480313584461926637256837089, −3.31874751359050490312370733802, −2.61237471789384733489531182439, −1.53647528102741022128099740579, −0.52408110145612652103271752356, 0.52408110145612652103271752356, 1.53647528102741022128099740579, 2.61237471789384733489531182439, 3.31874751359050490312370733802, 4.01480313584461926637256837089, 4.63021633725152443209733084264, 4.99207371310814885896269263171, 5.69886437390822823949011200716, 6.01612086119737147435067066575, 6.84899310638551761424185074277, 7.11431484901848047337285972916, 7.77531817530388024228130269937, 8.194417309724579054838462347537, 8.568822291342378708563372922224, 9.135302092333353652652186528333

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