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Properties

Label 2-40e2-80.27-c1-0-1
Degree $2$
Conductor $1600$
Sign $-0.999 + 0.00616i$
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  − 2.96·3-s + (−0.115 − 0.115i)7-s + 5.79·9-s + (−2.95 + 2.95i)11-s + 1.55i·13-s + (−0.299 − 0.299i)17-s + (2.26 − 2.26i)19-s + (0.341 + 0.341i)21-s + (4.14 − 4.14i)23-s − 8.28·27-s + (0.289 + 0.289i)29-s + 4.18i·31-s + (8.77 − 8.77i)33-s − 1.63i·37-s − 4.62i·39-s + ⋯
L(s)  = 1  − 1.71·3-s + (−0.0435 − 0.0435i)7-s + 1.93·9-s + (−0.892 + 0.892i)11-s + 0.432i·13-s + (−0.0726 − 0.0726i)17-s + (0.519 − 0.519i)19-s + (0.0744 + 0.0744i)21-s + (0.864 − 0.864i)23-s − 1.59·27-s + (0.0537 + 0.0537i)29-s + 0.751i·31-s + (1.52 − 1.52i)33-s − 0.269i·37-s − 0.739i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.999 + 0.00616i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.999 + 0.00616i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08988101224\)
\(L(\frac12)\) \(\approx\) \(0.08988101224\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.96T + 3T^{2} \)
7 \( 1 + (0.115 + 0.115i)T + 7iT^{2} \)
11 \( 1 + (2.95 - 2.95i)T - 11iT^{2} \)
13 \( 1 - 1.55iT - 13T^{2} \)
17 \( 1 + (0.299 + 0.299i)T + 17iT^{2} \)
19 \( 1 + (-2.26 + 2.26i)T - 19iT^{2} \)
23 \( 1 + (-4.14 + 4.14i)T - 23iT^{2} \)
29 \( 1 + (-0.289 - 0.289i)T + 29iT^{2} \)
31 \( 1 - 4.18iT - 31T^{2} \)
37 \( 1 + 1.63iT - 37T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 - 6.72iT - 43T^{2} \)
47 \( 1 + (-4.38 + 4.38i)T - 47iT^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + (1.63 + 1.63i)T + 59iT^{2} \)
61 \( 1 + (1.23 - 1.23i)T - 61iT^{2} \)
67 \( 1 - 2.49iT - 67T^{2} \)
71 \( 1 + 8.00T + 71T^{2} \)
73 \( 1 + (-1.12 - 1.12i)T + 73iT^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 - 1.62T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (9.69 + 9.69i)T + 97iT^{2} \)
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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

−10.02703881727834497060961179510, −9.244974457452427907989146603092, −8.050687986822648596872007835593, −7.03643864064221807621351472139, −6.67904951472861455723439645314, −5.62949771438556299822108099390, −4.91618362784684687970483989574, −4.39981355756538338370233452247, −2.81262224593353933832337758269, −1.35983389570333783824982735327, 0.05072058414013360982383670036, 1.25730562056835365001223712329, 2.93753295301080497727713457271, 4.11997040828548818663193620917, 5.27260658003815452131103571325, 5.57331428255510463135670015233, 6.35435274549526630515238087221, 7.32335120053849318699261756011, 8.000330811323064271844559976721, 9.176496025465059184371496224142

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