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Properties

Label 4-38160-1.1-c1e2-0-0
Degree $4$
Conductor $38160$
Sign $1$
Analytic cond. $2.43311$
Root an. cond. $1.24893$
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  − 2·4-s + 2·5-s + 9-s + 13-s + 4·16-s − 3·17-s − 4·20-s + 2·25-s + 15·29-s − 2·36-s + 7·37-s − 6·41-s + 2·45-s − 10·49-s − 2·52-s + 8·53-s + 7·61-s − 8·64-s + 2·65-s + 6·68-s + 4·73-s + 8·80-s + 81-s − 6·85-s − 5·97-s − 4·100-s + 3·101-s + ⋯
L(s)  = 1  − 4-s + 0.894·5-s + 1/3·9-s + 0.277·13-s + 16-s − 0.727·17-s − 0.894·20-s + 2/5·25-s + 2.78·29-s − 1/3·36-s + 1.15·37-s − 0.937·41-s + 0.298·45-s − 1.42·49-s − 0.277·52-s + 1.09·53-s + 0.896·61-s − 64-s + 0.248·65-s + 0.727·68-s + 0.468·73-s + 0.894·80-s + 1/9·81-s − 0.650·85-s − 0.507·97-s − 2/5·100-s + 0.298·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38160 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38160 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(38160\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 53\)
Sign: $1$
Analytic conductor: \(2.43311\)
Root analytic conductor: \(1.24893\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 38160,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.255955183\)
\(L(\frac12)\) \(\approx\) \(1.255955183\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 9 T + p T^{2} ) \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.ab_y
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.23.a_b
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ap_ei
31$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.31.a_bo
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.ah_co
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.43.a_ack
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
59$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.a_bl
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.ah_ek
67$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.67.a_cd
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.83.a_abp
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.a_abv
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.f_dm
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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

−10.18131788136392404774411022868, −9.810095700661265201713993318759, −9.374530046950493830795667280792, −8.708971189191415512153388224065, −8.390730796214772048848152189877, −7.88784920303591499958946815949, −6.95185108391462861156963704430, −6.49925377239135871879751355772, −5.97407511400644632798082386075, −5.20573683896492110609613442877, −4.71438036821825431331852022899, −4.16511504504355075124583820656, −3.24351485141889210681592335905, −2.37016822170639086166641345252, −1.13873598999310482201173531344, 1.13873598999310482201173531344, 2.37016822170639086166641345252, 3.24351485141889210681592335905, 4.16511504504355075124583820656, 4.71438036821825431331852022899, 5.20573683896492110609613442877, 5.97407511400644632798082386075, 6.49925377239135871879751355772, 6.95185108391462861156963704430, 7.88784920303591499958946815949, 8.390730796214772048848152189877, 8.708971189191415512153388224065, 9.374530046950493830795667280792, 9.810095700661265201713993318759, 10.18131788136392404774411022868

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