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Properties

Label 4-540800-1.1-c1e2-0-60
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  + 5-s + 9-s + 2·13-s − 8·17-s − 4·25-s − 8·29-s − 6·37-s + 8·41-s + 45-s − 5·49-s − 4·53-s + 16·61-s + 2·65-s + 4·73-s − 8·81-s − 8·85-s − 8·89-s + 8·97-s + 8·101-s − 4·109-s − 8·113-s + 2·117-s + 10·121-s − 9·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.447·5-s + 1/3·9-s + 0.554·13-s − 1.94·17-s − 4/5·25-s − 1.48·29-s − 0.986·37-s + 1.24·41-s + 0.149·45-s − 5/7·49-s − 0.549·53-s + 2.04·61-s + 0.248·65-s + 0.468·73-s − 8/9·81-s − 0.867·85-s − 0.847·89-s + 0.812·97-s + 0.796·101-s − 0.383·109-s − 0.752·113-s + 0.184·117-s + 0.909·121-s − 0.804·125-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$C_2$ \( 1 - T + p T^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.i_bp
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.19.a_as
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.i_cs
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.31.a_bi
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.g_bv
41$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.41.ai_bi
43$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.43.a_x
47$C_2^2$ \( 1 - 67 T^{2} + p^{2} T^{4} \) 2.47.a_acp
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.e_ec
59$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.59.a_abi
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.67.a_adi
71$C_2^2$ \( 1 + 123 T^{2} + p^{2} T^{4} \) 2.71.a_et
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ae_di
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.83.a_fe
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.i_dq
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.97.ai_hy
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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

−8.338553307682856225092429230097, −7.72401215098394316857960919575, −7.37449108338009914548985271184, −6.74384923569213766673040293169, −6.52934913265744683166783372535, −5.91688522438786436179622418389, −5.54240242707092290283292706737, −4.94372464004930938010012196924, −4.39841261775256014566678615225, −3.87678172706588344530134946452, −3.47479905070959796311599606877, −2.46169005466521813800561678226, −2.10713778030478682210019930684, −1.35721597720795742844838032124, 0, 1.35721597720795742844838032124, 2.10713778030478682210019930684, 2.46169005466521813800561678226, 3.47479905070959796311599606877, 3.87678172706588344530134946452, 4.39841261775256014566678615225, 4.94372464004930938010012196924, 5.54240242707092290283292706737, 5.91688522438786436179622418389, 6.52934913265744683166783372535, 6.74384923569213766673040293169, 7.37449108338009914548985271184, 7.72401215098394316857960919575, 8.338553307682856225092429230097

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