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Properties

Label 4-540800-1.1-c1e2-0-40
Degree $4$
Conductor $540800$
Sign $-1$
Analytic cond. $34.4818$
Root an. cond. $2.42324$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·9-s − 2·13-s + 8·17-s − 25-s − 8·29-s + 6·37-s − 2·41-s + 8·45-s + 10·49-s + 14·53-s − 24·61-s + 4·65-s + 6·73-s + 7·81-s − 16·85-s + 12·89-s + 12·97-s + 8·101-s − 14·109-s − 12·113-s + 8·117-s + 10·121-s + 12·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 4/3·9-s − 0.554·13-s + 1.94·17-s − 1/5·25-s − 1.48·29-s + 0.986·37-s − 0.312·41-s + 1.19·45-s + 10/7·49-s + 1.92·53-s − 3.07·61-s + 0.496·65-s + 0.702·73-s + 7/9·81-s − 1.73·85-s + 1.27·89-s + 1.21·97-s + 0.796·101-s − 1.34·109-s − 1.12·113-s + 0.739·117-s + 0.909·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540800\)    =    \(2^{7} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(34.4818\)
Root analytic conductor: \(2.42324\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 540800,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ai_bu
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.23.a_aq
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.i_cs
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.31.a_o
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ag_de
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.c_bi
43$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.43.a_i
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.ao_fq
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \) 2.67.a_aew
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.ag_ec
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.83.a_aec
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.am_gk
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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

−8.108471461957711940524755876100, −7.70849025586414781994228858349, −7.55832845306086563515684567516, −7.09541036819687202594285382903, −6.27793377533630052555580385370, −5.81022946833604319443960790425, −5.58253654987918468533561720633, −5.00681501106578448645058556654, −4.40827794638602250495511262690, −3.67471689592303987361200626806, −3.46466446117738800019021722984, −2.76996570675243436327098180739, −2.16973055198154386303552635037, −1.04651793970925095688712143243, 0, 1.04651793970925095688712143243, 2.16973055198154386303552635037, 2.76996570675243436327098180739, 3.46466446117738800019021722984, 3.67471689592303987361200626806, 4.40827794638602250495511262690, 5.00681501106578448645058556654, 5.58253654987918468533561720633, 5.81022946833604319443960790425, 6.27793377533630052555580385370, 7.09541036819687202594285382903, 7.55832845306086563515684567516, 7.70849025586414781994228858349, 8.108471461957711940524755876100

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