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Properties

Label 4-16160-1.1-c1e2-0-0
Degree $4$
Conductor $16160$
Sign $1$
Analytic cond. $1.03037$
Root an. cond. $1.00750$
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 − 2·9-s − 10-s + 4·13-s + 16-s + 6·17-s + 2·18-s + 20-s − 4·25-s − 4·26-s − 32-s − 6·34-s − 2·36-s + 4·37-s − 40-s + 12·41-s − 2·45-s + 2·49-s + 4·50-s + 4·52-s − 6·53-s − 8·61-s + 64-s + 4·65-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.471·18-s + 0.223·20-s − 4/5·25-s − 0.784·26-s − 0.176·32-s − 1.02·34-s − 1/3·36-s + 0.657·37-s − 0.158·40-s + 1.87·41-s − 0.298·45-s + 2/7·49-s + 0.565·50-s + 0.554·52-s − 0.824·53-s − 1.02·61-s + 1/8·64-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16160 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16160 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16160\)    =    \(2^{5} \cdot 5 \cdot 101\)
Sign: $1$
Analytic conductor: \(1.03037\)
Root analytic conductor: \(1.00750\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16160,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8817605090\)
\(L(\frac12)\) \(\approx\) \(0.8817605090\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
101$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 12 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
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$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.23.a_ao
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.43.a_cm
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.g_ec
59$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.59.a_q
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.i_s
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ag_gw
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.c_ek
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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

−10.82265763616651707502159365425, −10.66926049039533793241109773218, −9.848216526644615973052403098227, −9.427759456524466024796236936618, −8.997893576013490471164355017547, −8.187235662033446430487802557631, −7.928699199590608071406471652525, −7.25834637272793573862950477659, −6.37263401356939755593249749792, −5.85209121831263901072739405119, −5.52336969470970097948062440710, −4.32961980429212981604270406376, −3.43365095699241301167157141281, −2.61804205034226632863640848754, −1.33550120530394758648455520017, 1.33550120530394758648455520017, 2.61804205034226632863640848754, 3.43365095699241301167157141281, 4.32961980429212981604270406376, 5.52336969470970097948062440710, 5.85209121831263901072739405119, 6.37263401356939755593249749792, 7.25834637272793573862950477659, 7.928699199590608071406471652525, 8.187235662033446430487802557631, 8.997893576013490471164355017547, 9.427759456524466024796236936618, 9.848216526644615973052403098227, 10.66926049039533793241109773218, 10.82265763616651707502159365425

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