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Properties

Label 2-40e2-80.27-c1-0-9
Degree $2$
Conductor $1600$
Sign $0.998 - 0.0601i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.998 - 0.0601i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.998 - 0.0601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7365512724\)
\(L(\frac12)\) \(\approx\) \(0.7365512724\)
\(L(\frac{3}{2})\) 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

−9.825394349950385356903775358384, −8.399684673266049662806839551385, −7.83377421685623387483811816877, −6.85225180705845310353082655403, −6.15872444813367722323789872256, −5.20663351986880692784391283089, −4.77757799990595756374469953338, −3.43125008507967378685913485024, −2.30629163513042197690867704210, −0.63106692514641875541045950448, 0.59147495118884937464178803110, 2.31828083879746962379287952981, 3.37233434938330059954843232593, 4.66552684006002247603804642936, 5.37573131423335387416056553813, 6.10460735660789536054241518004, 6.74814636425899638961669469412, 7.74263825944156582081044364927, 8.760055490724277538974766876877, 9.300494446421993277253382913543

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