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Properties

Label 4-32800-1.1-c1e2-0-0
Degree $4$
Conductor $32800$
Sign $1$
Analytic cond. $2.09135$
Root an. cond. $1.20256$
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 − 8-s + 9-s + 4·13-s + 16-s − 3·17-s − 18-s + 25-s − 4·26-s + 12·29-s − 32-s + 3·34-s + 36-s + 37-s − 10·41-s + 8·49-s − 50-s + 4·52-s − 12·58-s + 19·61-s + 64-s − 3·68-s − 72-s − 14·73-s − 74-s − 8·81-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 1/5·25-s − 0.784·26-s + 2.22·29-s − 0.176·32-s + 0.514·34-s + 1/6·36-s + 0.164·37-s − 1.56·41-s + 8/7·49-s − 0.141·50-s + 0.554·52-s − 1.57·58-s + 2.43·61-s + 1/8·64-s − 0.363·68-s − 0.117·72-s − 1.63·73-s − 0.116·74-s − 8/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32800\)    =    \(2^{5} \cdot 5^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(2.09135\)
Root analytic conductor: \(1.20256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32800,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9770161319\)
\(L(\frac12)\) \(\approx\) \(0.9770161319\)
\(L(\frac{3}{2})\) 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_1$ \( ( 1 - T )( 1 + T ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 9 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.11.a_b
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.ae_v
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.23.a_af
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.29.am_dh
31$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \) 2.31.a_al
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.ab_s
43$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.43.a_e
47$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \) 2.47.a_cy
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.59.a_ca
61$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.61.at_ic
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.a_abj
71$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.71.a_abs
73$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.o_he
79$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \) 2.79.a_aey
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.am_cs
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.ak_ic
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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

−10.35498112092973149441903747321, −10.07008029297074072933874945246, −9.349713225158248770872181312846, −8.682965540232716608884187813243, −8.528157083526384374506743356228, −7.968214831903292021745382287133, −7.12104180125266944199402815074, −6.74171936786759923382254977561, −6.24755174977719468758829417548, −5.51739895580156761712235978379, −4.72651594216830803557903319860, −4.02796523216584482489034444874, −3.19313491633883166277182328045, −2.29161703069160965400771490022, −1.13584077747014033881186326670, 1.13584077747014033881186326670, 2.29161703069160965400771490022, 3.19313491633883166277182328045, 4.02796523216584482489034444874, 4.72651594216830803557903319860, 5.51739895580156761712235978379, 6.24755174977719468758829417548, 6.74171936786759923382254977561, 7.12104180125266944199402815074, 7.968214831903292021745382287133, 8.528157083526384374506743356228, 8.682965540232716608884187813243, 9.349713225158248770872181312846, 10.07008029297074072933874945246, 10.35498112092973149441903747321

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