Properties

Label 4-320e2-1.1-c1e2-0-13
Degree $4$
Conductor $102400$
Sign $1$
Analytic cond. $6.52911$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 2·9-s − 8·11-s + 8·19-s − 25-s + 4·29-s + 4·41-s + 4·45-s + 10·49-s − 16·55-s + 24·59-s + 20·61-s − 16·71-s + 32·79-s − 5·81-s − 12·89-s + 16·95-s − 16·99-s − 12·101-s − 12·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯
L(s)  = 1  + 0.894·5-s + 2/3·9-s − 2.41·11-s + 1.83·19-s − 1/5·25-s + 0.742·29-s + 0.624·41-s + 0.596·45-s + 10/7·49-s − 2.15·55-s + 3.12·59-s + 2.56·61-s − 1.89·71-s + 3.60·79-s − 5/9·81-s − 1.27·89-s + 1.64·95-s − 1.60·99-s − 1.19·101-s − 1.14·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(102400\)    =    \(2^{12} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(6.52911\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 102400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722474975\)
\(L(\frac12)\) \(\approx\) \(1.722474975\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−11.70788594371610619857230734194, −11.56003040110246279599858175777, −10.67417227178092479248836645418, −10.35601431790166315181169347920, −10.18685347995366398591103840290, −9.442278828225751355671888691640, −9.439291448262045443523774789712, −8.286652205325374798821505754513, −8.244666969700496587120338990414, −7.49390492065617882517364478080, −7.16092932639731528465211978098, −6.61140314742272237450934732165, −5.67291059682368933672061423358, −5.41765569838809309218604951719, −5.15679754884998928140906335902, −4.27704939201964139077046919386, −3.51247017092291311420608862381, −2.57721743382362538607804729472, −2.33249497133181333909952594374, −1.00408633879289146220868726536, 1.00408633879289146220868726536, 2.33249497133181333909952594374, 2.57721743382362538607804729472, 3.51247017092291311420608862381, 4.27704939201964139077046919386, 5.15679754884998928140906335902, 5.41765569838809309218604951719, 5.67291059682368933672061423358, 6.61140314742272237450934732165, 7.16092932639731528465211978098, 7.49390492065617882517364478080, 8.244666969700496587120338990414, 8.286652205325374798821505754513, 9.439291448262045443523774789712, 9.442278828225751355671888691640, 10.18685347995366398591103840290, 10.35601431790166315181169347920, 10.67417227178092479248836645418, 11.56003040110246279599858175777, 11.70788594371610619857230734194

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