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Properties

Label 2-40e2-80.3-c1-0-33
Degree $2$
Conductor $1600$
Sign $-0.258 + 0.965i$
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  + 1.93·3-s + (0.896 − 0.896i)7-s + 0.732·9-s + (−4.09 − 4.09i)11-s − 4.89i·13-s + (−0.707 + 0.707i)17-s + (−3.09 − 3.09i)19-s + (1.73 − 1.73i)21-s + (−2.96 − 2.96i)23-s − 4.38·27-s + (1.26 − 1.26i)29-s + 4.19i·31-s + (−7.91 − 7.91i)33-s + 10.9i·37-s − 9.46i·39-s + ⋯
L(s)  = 1  + 1.11·3-s + (0.338 − 0.338i)7-s + 0.244·9-s + (−1.23 − 1.23i)11-s − 1.35i·13-s + (−0.171 + 0.171i)17-s + (−0.710 − 0.710i)19-s + (0.377 − 0.377i)21-s + (−0.618 − 0.618i)23-s − 0.843·27-s + (0.235 − 0.235i)29-s + 0.753i·31-s + (−1.37 − 1.37i)33-s + 1.79i·37-s − 1.51i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.778192071\)
\(L(\frac12)\) \(\approx\) \(1.778192071\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.93T + 3T^{2} \)
7 \( 1 + (-0.896 + 0.896i)T - 7iT^{2} \)
11 \( 1 + (4.09 + 4.09i)T + 11iT^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 + (0.707 - 0.707i)T - 17iT^{2} \)
19 \( 1 + (3.09 + 3.09i)T + 19iT^{2} \)
23 \( 1 + (2.96 + 2.96i)T + 23iT^{2} \)
29 \( 1 + (-1.26 + 1.26i)T - 29iT^{2} \)
31 \( 1 - 4.19iT - 31T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 + 9.14iT - 43T^{2} \)
47 \( 1 + (1.41 + 1.41i)T + 47iT^{2} \)
53 \( 1 - 9.89T + 53T^{2} \)
59 \( 1 + (-4.26 + 4.26i)T - 59iT^{2} \)
61 \( 1 + (-7.19 - 7.19i)T + 61iT^{2} \)
67 \( 1 - 1.55iT - 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + (3.91 - 3.91i)T - 73iT^{2} \)
79 \( 1 - 8.19T + 79T^{2} \)
83 \( 1 + 4.65T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + (3.10 - 3.10i)T - 97iT^{2} \)
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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

−8.797323168004684691767580310930, −8.357665777176228704289685327117, −7.960575191528896099857329219087, −6.94495958603525378730658366452, −5.80431062874122893271608290347, −5.07766846513063739957872445925, −3.85315064015299485322241114919, −2.98598562203120617096646565696, −2.32660112943375842547534645310, −0.54349729438953045297632830117, 2.04481842053426536617129564626, 2.30420628430836343557241715828, 3.69866851939717139231265497848, 4.50427527870242357558114073304, 5.45849378567849226407670686123, 6.54255003417300099390867826187, 7.57167278795852903535708217173, 7.988402389366407676198637848956, 8.855933457500168005875814121351, 9.528548158092302859933830545915

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