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Properties

Label 2-40e2-80.29-c1-0-11
Degree $2$
Conductor $1600$
Sign $0.573 - 0.819i$
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  + (−1.82 + 1.82i)3-s − 4.50·7-s − 3.68i·9-s + (1.64 − 1.64i)11-s + (1.51 − 1.51i)13-s + 1.45i·17-s + (−2.67 − 2.67i)19-s + (8.24 − 8.24i)21-s + 2.37·23-s + (1.24 + 1.24i)27-s + (−0.924 − 0.924i)29-s + 7.20·31-s + 5.99i·33-s + (−5.21 − 5.21i)37-s + 5.55i·39-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)3-s − 1.70·7-s − 1.22i·9-s + (0.494 − 0.494i)11-s + (0.421 − 0.421i)13-s + 0.353i·17-s + (−0.614 − 0.614i)19-s + (1.79 − 1.79i)21-s + 0.495·23-s + (0.239 + 0.239i)27-s + (−0.171 − 0.171i)29-s + 1.29·31-s + 1.04i·33-s + (−0.856 − 0.856i)37-s + 0.888i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.573 - 0.819i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.573 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7643632046\)
\(L(\frac12)\) \(\approx\) \(0.7643632046\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.82 - 1.82i)T - 3iT^{2} \)
7 \( 1 + 4.50T + 7T^{2} \)
11 \( 1 + (-1.64 + 1.64i)T - 11iT^{2} \)
13 \( 1 + (-1.51 + 1.51i)T - 13iT^{2} \)
17 \( 1 - 1.45iT - 17T^{2} \)
19 \( 1 + (2.67 + 2.67i)T + 19iT^{2} \)
23 \( 1 - 2.37T + 23T^{2} \)
29 \( 1 + (0.924 + 0.924i)T + 29iT^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 + (5.21 + 5.21i)T + 37iT^{2} \)
41 \( 1 - 6.41iT - 41T^{2} \)
43 \( 1 + (-7.65 - 7.65i)T + 43iT^{2} \)
47 \( 1 - 2.51iT - 47T^{2} \)
53 \( 1 + (1.50 + 1.50i)T + 53iT^{2} \)
59 \( 1 + (5.31 - 5.31i)T - 59iT^{2} \)
61 \( 1 + (1.02 + 1.02i)T + 61iT^{2} \)
67 \( 1 + (-5.22 + 5.22i)T - 67iT^{2} \)
71 \( 1 + 1.92iT - 71T^{2} \)
73 \( 1 - 1.39T + 73T^{2} \)
79 \( 1 - 5.06T + 79T^{2} \)
83 \( 1 + (-2.44 + 2.44i)T - 83iT^{2} \)
89 \( 1 - 9.36iT - 89T^{2} \)
97 \( 1 - 18.6iT - 97T^{2} \)
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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

−9.461435926555777412230467955013, −9.209036178164681787447818906876, −8.017735497106358589352457015489, −6.61769262475139899703899679275, −6.30206343871476348173061473929, −5.53975178140953455948795158498, −4.48681941804202980531304772081, −3.70041176925018329065213414040, −2.85300814430980835055608533933, −0.67140758519375132701755630135, 0.59524160032132194741431106093, 1.89850365964226741897079190692, 3.20605260771720063375030581123, 4.25934911527482537452105170932, 5.52491112060008793000624655691, 6.22913768132318444207931112454, 6.80349638386959233412966066234, 7.22987013586587618300511137588, 8.501243230150371363435500113852, 9.349951000572851752807697385943

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