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Properties

Label 2-40e2-80.67-c1-0-15
Degree $2$
Conductor $1600$
Sign $0.886 + 0.461i$
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  − 0.692i·3-s + (−0.343 − 0.343i)7-s + 2.52·9-s + (−0.843 − 0.843i)11-s + 3.68·13-s + (−0.412 − 0.412i)17-s + (5.37 + 5.37i)19-s + (−0.238 + 0.238i)21-s + (−3.08 + 3.08i)23-s − 3.82i·27-s + (−4.22 + 4.22i)29-s − 8.75i·31-s + (−0.584 + 0.584i)33-s + 5.41·37-s − 2.55i·39-s + ⋯
L(s)  = 1  − 0.399i·3-s + (−0.129 − 0.129i)7-s + 0.840·9-s + (−0.254 − 0.254i)11-s + 1.02·13-s + (−0.0999 − 0.0999i)17-s + (1.23 + 1.23i)19-s + (−0.0519 + 0.0519i)21-s + (−0.643 + 0.643i)23-s − 0.735i·27-s + (−0.785 + 0.785i)29-s − 1.57i·31-s + (−0.101 + 0.101i)33-s + 0.890·37-s − 0.408i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.886 + 0.461i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.886 + 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.896673083\)
\(L(\frac12)\) \(\approx\) \(1.896673083\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.692iT - 3T^{2} \)
7 \( 1 + (0.343 + 0.343i)T + 7iT^{2} \)
11 \( 1 + (0.843 + 0.843i)T + 11iT^{2} \)
13 \( 1 - 3.68T + 13T^{2} \)
17 \( 1 + (0.412 + 0.412i)T + 17iT^{2} \)
19 \( 1 + (-5.37 - 5.37i)T + 19iT^{2} \)
23 \( 1 + (3.08 - 3.08i)T - 23iT^{2} \)
29 \( 1 + (4.22 - 4.22i)T - 29iT^{2} \)
31 \( 1 + 8.75iT - 31T^{2} \)
37 \( 1 - 5.41T + 37T^{2} \)
41 \( 1 - 2.54iT - 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 + (-4.56 + 4.56i)T - 47iT^{2} \)
53 \( 1 - 6.07iT - 53T^{2} \)
59 \( 1 + (-7.33 + 7.33i)T - 59iT^{2} \)
61 \( 1 + (4.81 + 4.81i)T + 61iT^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 + (-6.87 - 6.87i)T + 73iT^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 7.15iT - 83T^{2} \)
89 \( 1 - 1.10T + 89T^{2} \)
97 \( 1 + (7.15 + 7.15i)T + 97iT^{2} \)
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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

−9.581720252381019373479023995718, −8.360717694480941520136385527483, −7.74195671439976740322113082489, −7.05148783031312111210024230709, −6.04677164761382693018515869041, −5.45046914127720275385540602689, −4.09815150386235052715381217740, −3.48999428730090597611649198065, −2.04967074703737858904163789700, −0.987279039827352664695577225187, 1.07310248424118046092319706297, 2.46816618246328311490757241860, 3.60069944940696896124452854351, 4.42020134770037026649926837000, 5.28128286160898010770862451809, 6.23570559778222217303628522305, 7.10346793375368751067326880217, 7.82626609783224272589539210643, 8.867660879475176408255122682362, 9.429466200202781728269825148622

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