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Properties

Label 4-20e2-1.1-c41e2-0-0
Degree $4$
Conductor $400$
Sign $1$
Analytic cond. $45344.8$
Root an. cond. $14.5925$
Motivic weight $41$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09e6·2-s + 2.19e12·4-s − 4.21e14·5-s + 8.83e20·10-s − 1.88e23·13-s − 4.83e24·16-s − 7.83e24·17-s − 9.25e26·20-s + 1.31e29·25-s + 3.94e29·26-s + 1.01e31·32-s + 1.64e31·34-s − 3.80e32·37-s − 2.64e33·41-s − 2.76e35·50-s − 4.14e35·52-s − 2.28e35·53-s − 1.55e37·61-s − 1.06e37·64-s + 7.92e37·65-s − 1.72e37·68-s − 9.43e37·73-s + 7.97e38·74-s + 2.03e39·80-s − 1.33e39·81-s + 5.54e39·82-s + 3.29e39·85-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.97·5-s + 2.79·10-s − 2.74·13-s − 16-s − 0.467·17-s − 1.97·20-s + 2.89·25-s + 3.88·26-s + 1.41·32-s + 0.661·34-s − 2.70·37-s − 2.29·41-s − 4.09·50-s − 2.74·52-s − 1.02·53-s − 3.91·61-s − 64-s + 5.42·65-s − 0.467·68-s − 0.597·73-s + 3.82·74-s + 1.97·80-s − 81-s + 3.24·82-s + 0.923·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+41/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(45344.8\)
Root analytic conductor: \(14.5925\)
Motivic weight: \(41\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 400,\ (\ :41/2, 41/2),\ 1)\)

Particular Values

\(L(21)\) \(\approx\) \(0.1033630176\)
\(L(\frac12)\) \(\approx\) \(0.1033630176\)
\(L(\frac{43}{2})\) 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)$
bad2$C_2$ \( 1 + p^{21} T + p^{41} T^{2} \)
5$C_2$ \( 1 + 421048874458084 T + p^{41} T^{2} \)
good3$C_2^2$ \( 1 + p^{82} T^{4} \)
7$C_2^2$ \( 1 + p^{82} T^{4} \)
11$C_2$ \( ( 1 - p^{41} T^{2} )^{2} \)
13$C_2$ \( ( 1 + \)\(71\!\cdots\!06\)\( T + p^{41} T^{2} )( 1 + \)\(11\!\cdots\!96\)\( T + p^{41} T^{2} ) \)
17$C_2$ \( ( 1 - \)\(19\!\cdots\!98\)\( T + p^{41} T^{2} )( 1 + \)\(27\!\cdots\!92\)\( T + p^{41} T^{2} ) \)
19$C_2$ \( ( 1 + p^{41} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{82} T^{4} \)
29$C_2$ \( ( 1 - \)\(18\!\cdots\!90\)\( T + p^{41} T^{2} )( 1 + \)\(18\!\cdots\!90\)\( T + p^{41} T^{2} ) \)
31$C_2$ \( ( 1 - p^{41} T^{2} )^{2} \)
37$C_2$ \( ( 1 + \)\(13\!\cdots\!02\)\( T + p^{41} T^{2} )( 1 + \)\(24\!\cdots\!12\)\( T + p^{41} T^{2} ) \)
41$C_2$ \( ( 1 + \)\(13\!\cdots\!08\)\( T + p^{41} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{82} T^{4} \)
47$C_2^2$ \( 1 + p^{82} T^{4} \)
53$C_2$ \( ( 1 - \)\(17\!\cdots\!14\)\( T + p^{41} T^{2} )( 1 + \)\(40\!\cdots\!96\)\( T + p^{41} T^{2} ) \)
59$C_2$ \( ( 1 + p^{41} T^{2} )^{2} \)
61$C_2$ \( ( 1 + \)\(77\!\cdots\!88\)\( T + p^{41} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{82} T^{4} \)
71$C_2$ \( ( 1 - p^{41} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(17\!\cdots\!84\)\( T + p^{41} T^{2} )( 1 + \)\(26\!\cdots\!06\)\( T + p^{41} T^{2} ) \)
79$C_2$ \( ( 1 + p^{41} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{82} T^{4} \)
89$C_2$ \( ( 1 - \)\(14\!\cdots\!90\)\( T + p^{41} T^{2} )( 1 + \)\(14\!\cdots\!90\)\( T + p^{41} T^{2} ) \)
97$C_2$ \( ( 1 + \)\(14\!\cdots\!82\)\( T + p^{41} T^{2} )( 1 + \)\(10\!\cdots\!92\)\( T + p^{41} T^{2} ) \)
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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

−10.88831877357609419781578056834, −10.80506330190273086604643162003, −9.976711369540928973420822366280, −9.569118824604693045680369914725, −8.794072388740840662846193771897, −8.459344863847172369065707172015, −7.81973153450016617134434376038, −7.44791841851448501267139703248, −6.93700447196332516180747313498, −6.75861449105334908635016086426, −5.34852930057208041983205488733, −4.66573618643595283655169259180, −4.63684025873281714407457932572, −3.75488155062864502324275923823, −2.98198411741780824587347488563, −2.72425124282518022898200658702, −1.66598155891349347235294039915, −1.52403772104437676493777238956, −0.29117649496110254010023504809, −0.23477296562998598846664271604, 0.23477296562998598846664271604, 0.29117649496110254010023504809, 1.52403772104437676493777238956, 1.66598155891349347235294039915, 2.72425124282518022898200658702, 2.98198411741780824587347488563, 3.75488155062864502324275923823, 4.63684025873281714407457932572, 4.66573618643595283655169259180, 5.34852930057208041983205488733, 6.75861449105334908635016086426, 6.93700447196332516180747313498, 7.44791841851448501267139703248, 7.81973153450016617134434376038, 8.459344863847172369065707172015, 8.794072388740840662846193771897, 9.569118824604693045680369914725, 9.976711369540928973420822366280, 10.80506330190273086604643162003, 10.88831877357609419781578056834

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