Properties

Label 2-40e2-80.43-c1-0-27
Degree $2$
Conductor $1600$
Sign $0.251 + 0.967i$
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  − 0.496i·3-s + (1.55 − 1.55i)7-s + 2.75·9-s + (4.19 − 4.19i)11-s − 5.09·13-s + (−0.213 + 0.213i)17-s + (0.844 − 0.844i)19-s + (−0.771 − 0.771i)21-s + (1.70 + 1.70i)23-s − 2.85i·27-s + (−2.24 − 2.24i)29-s − 0.818i·31-s + (−2.08 − 2.08i)33-s + 5.12·37-s + 2.52i·39-s + ⋯
L(s)  = 1  − 0.286i·3-s + (0.587 − 0.587i)7-s + 0.917·9-s + (1.26 − 1.26i)11-s − 1.41·13-s + (−0.0517 + 0.0517i)17-s + (0.193 − 0.193i)19-s + (−0.168 − 0.168i)21-s + (0.356 + 0.356i)23-s − 0.549i·27-s + (−0.417 − 0.417i)29-s − 0.146i·31-s + (−0.362 − 0.362i)33-s + 0.842·37-s + 0.405i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.251 + 0.967i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.919314884\)
\(L(\frac12)\) \(\approx\) \(1.919314884\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.496iT - 3T^{2} \)
7 \( 1 + (-1.55 + 1.55i)T - 7iT^{2} \)
11 \( 1 + (-4.19 + 4.19i)T - 11iT^{2} \)
13 \( 1 + 5.09T + 13T^{2} \)
17 \( 1 + (0.213 - 0.213i)T - 17iT^{2} \)
19 \( 1 + (-0.844 + 0.844i)T - 19iT^{2} \)
23 \( 1 + (-1.70 - 1.70i)T + 23iT^{2} \)
29 \( 1 + (2.24 + 2.24i)T + 29iT^{2} \)
31 \( 1 + 0.818iT - 31T^{2} \)
37 \( 1 - 5.12T + 37T^{2} \)
41 \( 1 - 3.34iT - 41T^{2} \)
43 \( 1 + 4.49T + 43T^{2} \)
47 \( 1 + (4.29 + 4.29i)T + 47iT^{2} \)
53 \( 1 - 1.00iT - 53T^{2} \)
59 \( 1 + (7.65 + 7.65i)T + 59iT^{2} \)
61 \( 1 + (1.90 - 1.90i)T - 61iT^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-2.70 + 2.70i)T - 73iT^{2} \)
79 \( 1 + 8.32T + 79T^{2} \)
83 \( 1 + 9.17iT - 83T^{2} \)
89 \( 1 - 4.25T + 89T^{2} \)
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

−9.398020853623668543994509186286, −8.319715942104304778557479858127, −7.59550879324257942776412597398, −6.91603404654721439417392822229, −6.13941083428047782651137410442, −4.97910504041084238095933136940, −4.23444746586698344946182033373, −3.27111892713829404645333442078, −1.87036757348077522022132255839, −0.809094907733246325425952033358, 1.48502935106093289378085196883, 2.38428584122542392313113492356, 3.81999831427550535110704490116, 4.67360578846916660752545902272, 5.17091578169678986904384770020, 6.53482843633452738066720481228, 7.15796214415866956138334876319, 7.88167387088801931801530108313, 9.020842398794356700010327966079, 9.584701427508293071453734419168

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