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Properties

Label 2-40e2-80.27-c1-0-20
Degree $2$
Conductor $1600$
Sign $0.982 - 0.184i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.85·3-s + (−0.458 − 0.458i)7-s + 5.15·9-s + (0.492 − 0.492i)11-s + 4.52i·13-s + (3.12 + 3.12i)17-s + (4.04 − 4.04i)19-s + (−1.31 − 1.31i)21-s + (−1.80 + 1.80i)23-s + 6.15·27-s + (3.83 + 3.83i)29-s + 0.139i·31-s + (1.40 − 1.40i)33-s − 5.84i·37-s + 12.9i·39-s + ⋯
L(s)  = 1  + 1.64·3-s + (−0.173 − 0.173i)7-s + 1.71·9-s + (0.148 − 0.148i)11-s + 1.25i·13-s + (0.758 + 0.758i)17-s + (0.928 − 0.928i)19-s + (−0.285 − 0.285i)21-s + (−0.376 + 0.376i)23-s + 1.18·27-s + (0.712 + 0.712i)29-s + 0.0251i·31-s + (0.244 − 0.244i)33-s − 0.960i·37-s + 2.06i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.982 - 0.184i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.982 - 0.184i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.186689459\)
\(L(\frac12)\) \(\approx\) \(3.186689459\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.85T + 3T^{2} \)
7 \( 1 + (0.458 + 0.458i)T + 7iT^{2} \)
11 \( 1 + (-0.492 + 0.492i)T - 11iT^{2} \)
13 \( 1 - 4.52iT - 13T^{2} \)
17 \( 1 + (-3.12 - 3.12i)T + 17iT^{2} \)
19 \( 1 + (-4.04 + 4.04i)T - 19iT^{2} \)
23 \( 1 + (1.80 - 1.80i)T - 23iT^{2} \)
29 \( 1 + (-3.83 - 3.83i)T + 29iT^{2} \)
31 \( 1 - 0.139iT - 31T^{2} \)
37 \( 1 + 5.84iT - 37T^{2} \)
41 \( 1 + 4.55iT - 41T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + (-4.14 + 4.14i)T - 47iT^{2} \)
53 \( 1 + 2.75T + 53T^{2} \)
59 \( 1 + (3.62 + 3.62i)T + 59iT^{2} \)
61 \( 1 + (-3.72 + 3.72i)T - 61iT^{2} \)
67 \( 1 - 3.32iT - 67T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + (2.55 + 2.55i)T + 73iT^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 - 3.35T + 89T^{2} \)
97 \( 1 + (-4.95 - 4.95i)T + 97iT^{2} \)
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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

−9.284464619818303733448306474607, −8.747479327875271222745680862885, −7.913594280696524667981313298033, −7.23678043749702903275956540660, −6.46089623255296592458526217664, −5.16319158633159938380169121184, −4.03710526206966016644916036384, −3.44222415174771492478971835317, −2.45024895598858243871767855797, −1.41784942990035915605035518910, 1.22997931052151010256538544286, 2.62455039685369728317992085052, 3.14309539247383655903540308131, 4.05507673436253443093207363732, 5.20087829668029166016618116244, 6.18997646638966129564191359444, 7.40296837329276370189459902347, 7.88133597815718118816435886570, 8.474178824396084034869473725228, 9.425096984445364777982804903063

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