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Properties

Label 2-40e2-80.29-c1-0-3
Degree $2$
Conductor $1600$
Sign $-0.692 - 0.721i$
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  + (−0.0623 + 0.0623i)3-s + 0.375·7-s + 2.99i·9-s + (−2.36 + 2.36i)11-s + (1.76 − 1.76i)13-s + 4.64i·17-s + (−2.34 − 2.34i)19-s + (−0.0234 + 0.0234i)21-s − 2.07·23-s + (−0.373 − 0.373i)27-s + (−2.55 − 2.55i)29-s − 8.51·31-s − 0.295i·33-s + (7.62 + 7.62i)37-s + 0.219i·39-s + ⋯
L(s)  = 1  + (−0.0359 + 0.0359i)3-s + 0.142·7-s + 0.997i·9-s + (−0.713 + 0.713i)11-s + (0.489 − 0.489i)13-s + 1.12i·17-s + (−0.539 − 0.539i)19-s + (−0.00511 + 0.00511i)21-s − 0.433·23-s + (−0.0718 − 0.0718i)27-s + (−0.474 − 0.474i)29-s − 1.52·31-s − 0.0513i·33-s + (1.25 + 1.25i)37-s + 0.0352i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.692 - 0.721i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.692 - 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8812974406\)
\(L(\frac12)\) \(\approx\) \(0.8812974406\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0623 - 0.0623i)T - 3iT^{2} \)
7 \( 1 - 0.375T + 7T^{2} \)
11 \( 1 + (2.36 - 2.36i)T - 11iT^{2} \)
13 \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \)
17 \( 1 - 4.64iT - 17T^{2} \)
19 \( 1 + (2.34 + 2.34i)T + 19iT^{2} \)
23 \( 1 + 2.07T + 23T^{2} \)
29 \( 1 + (2.55 + 2.55i)T + 29iT^{2} \)
31 \( 1 + 8.51T + 31T^{2} \)
37 \( 1 + (-7.62 - 7.62i)T + 37iT^{2} \)
41 \( 1 + 3.77iT - 41T^{2} \)
43 \( 1 + (6.21 + 6.21i)T + 43iT^{2} \)
47 \( 1 - 9.71iT - 47T^{2} \)
53 \( 1 + (-3.03 - 3.03i)T + 53iT^{2} \)
59 \( 1 + (8.11 - 8.11i)T - 59iT^{2} \)
61 \( 1 + (-0.728 - 0.728i)T + 61iT^{2} \)
67 \( 1 + (-0.969 + 0.969i)T - 67iT^{2} \)
71 \( 1 - 9.14iT - 71T^{2} \)
73 \( 1 + 7.56T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 - 3.86iT - 97T^{2} \)
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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

−9.806371969777221788344632513398, −8.819091917547217943686351898853, −7.976107145828082053226978631818, −7.56646952036393378770799312295, −6.41276242863080350963337318494, −5.57115074292113539074193800378, −4.76544171859033028694630907036, −3.88381514461239486699819480273, −2.59847370804366037395370697409, −1.67679405525534112492013919951, 0.32596795113944590765760699154, 1.82728041950513395187189407587, 3.14661491507752891996186707993, 3.89877600476203573183379014607, 5.04137127904529811281674820273, 5.90674929882876868901015144136, 6.61995987102659503291635925398, 7.55357962040500710837258895349, 8.340916127421210151988221333925, 9.186397870595526047497770406951

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