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Properties

Label 2-40e2-4.3-c0-0-0
Degree $2$
Conductor $1600$
Sign $1$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s + 2·29-s − 2·41-s + 49-s + 2·61-s + 81-s − 2·89-s − 2·101-s − 2·109-s + ⋯
L(s)  = 1  + 9-s + 2·29-s − 2·41-s + 49-s + 2·61-s + 81-s − 2·89-s − 2·101-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1600} (1151, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.213766981\)
\(L(\frac12)\) \(\approx\) \(1.213766981\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 + T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 + T )^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.807521255160318372723691246395, −8.712022103797310801500284099304, −8.103959676048039396409050848934, −7.04369177112671738016281878840, −6.59732373920539661883058621180, −5.42458642660661035278624521338, −4.59188543107590803390289081321, −3.73028496695088311683965279716, −2.56279133550929663848643454325, −1.29071805040217758490552346841, 1.29071805040217758490552346841, 2.56279133550929663848643454325, 3.73028496695088311683965279716, 4.59188543107590803390289081321, 5.42458642660661035278624521338, 6.59732373920539661883058621180, 7.04369177112671738016281878840, 8.103959676048039396409050848934, 8.712022103797310801500284099304, 9.807521255160318372723691246395

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