Properties

Label 4-168480-1.1-c1e2-0-1
Degree $4$
Conductor $168480$
Sign $1$
Analytic cond. $10.7424$
Root an. cond. $1.81040$
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 + 4-s + 5-s + 8-s + 10-s − 3·13-s + 16-s + 6·17-s + 20-s − 4·25-s − 3·26-s + 6·29-s + 32-s + 6·34-s + 4·37-s + 40-s + 14·49-s − 4·50-s − 3·52-s + 6·58-s − 2·61-s + 64-s − 3·65-s + 6·68-s + 4·73-s + 4·74-s + 80-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.832·13-s + 1/4·16-s + 1.45·17-s + 0.223·20-s − 4/5·25-s − 0.588·26-s + 1.11·29-s + 0.176·32-s + 1.02·34-s + 0.657·37-s + 0.158·40-s + 2·49-s − 0.565·50-s − 0.416·52-s + 0.787·58-s − 0.256·61-s + 1/8·64-s − 0.372·65-s + 0.727·68-s + 0.468·73-s + 0.464·74-s + 0.111·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168480 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168480 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(168480\)    =    \(2^{5} \cdot 3^{4} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(10.7424\)
Root analytic conductor: \(1.81040\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 168480,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.882179168\)
\(L(\frac12)\) \(\approx\) \(2.882179168\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
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$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.43.a_cy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.c_bq
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.g_ec
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−9.273929926966407120703243088013, −8.763077774724861816648258036874, −8.103057818740215840301437151271, −7.64145340943707910190665727819, −7.33351741478492979478258572485, −6.67924072818607010840500599451, −6.09962680495509908804233627574, −5.73695515141950226267871118174, −5.18149554716019271725491985200, −4.70579199748302297626620176053, −4.03144326619956465039150001297, −3.42856705058392951499862656907, −2.70583544879978146447085120520, −2.15208911038553247046831385073, −1.07337040755710952743210609152, 1.07337040755710952743210609152, 2.15208911038553247046831385073, 2.70583544879978146447085120520, 3.42856705058392951499862656907, 4.03144326619956465039150001297, 4.70579199748302297626620176053, 5.18149554716019271725491985200, 5.73695515141950226267871118174, 6.09962680495509908804233627574, 6.67924072818607010840500599451, 7.33351741478492979478258572485, 7.64145340943707910190665727819, 8.103057818740215840301437151271, 8.763077774724861816648258036874, 9.273929926966407120703243088013

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