Properties

Label 2-544-17.16-c1-0-5
Degree $2$
Conductor $544$
Sign $0.443 - 0.896i$
Analytic cond. $4.34386$
Root an. cond. $2.08419$
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  + 1.08i·3-s − 1.53i·5-s + 2.61i·7-s + 1.82·9-s + 1.08i·11-s − 0.828·13-s + 1.65·15-s + (−1.82 + 3.69i)17-s + 5.65·19-s − 2.82·21-s + 2.61i·23-s + 2.65·25-s + 5.22i·27-s − 5.86i·29-s + 5.67i·31-s + ⋯
L(s)  = 1  + 0.624i·3-s − 0.684i·5-s + 0.987i·7-s + 0.609·9-s + 0.326i·11-s − 0.229·13-s + 0.427·15-s + (−0.443 + 0.896i)17-s + 1.29·19-s − 0.617·21-s + 0.544i·23-s + 0.531·25-s + 1.00i·27-s − 1.08i·29-s + 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25944 + 0.782033i\)
\(L(\frac12)\) \(\approx\) \(1.25944 + 0.782033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (1.82 - 3.69i)T \)
good3 \( 1 - 1.08iT - 3T^{2} \)
5 \( 1 + 1.53iT - 5T^{2} \)
7 \( 1 - 2.61iT - 7T^{2} \)
11 \( 1 - 1.08iT - 11T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 - 2.61iT - 23T^{2} \)
29 \( 1 + 5.86iT - 29T^{2} \)
31 \( 1 - 5.67iT - 31T^{2} \)
37 \( 1 - 1.53iT - 37T^{2} \)
41 \( 1 - 7.39iT - 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 9.65T + 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 - 2.34T + 67T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 4.32iT - 73T^{2} \)
79 \( 1 + 7.83iT - 79T^{2} \)
83 \( 1 + 9.65T + 83T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−10.91158923844998000803591936823, −9.859728662922451673337299977639, −9.335789743087197409838587415606, −8.481535686186088457121962124841, −7.48208248945253761014711568636, −6.24699725763039147733763689977, −5.16997862292420710193679427085, −4.50503984309814059596360863998, −3.20430732083412609744253608849, −1.63235015452863972467470803121, 0.967469534409172895191270237335, 2.59727287743824178564811242784, 3.81454927537783687297198900722, 4.98859549361015215707632068272, 6.34759342387394003935191585598, 7.30996003741557697457753194959, 7.43455263416641366463514353952, 8.899723650984458629380321486509, 9.928462115475867337596883049325, 10.65210344894995167640381771817

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