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Properties

Label 2-544-136.101-c1-0-13
Degree $2$
Conductor $544$
Sign $0.908 + 0.418i$
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  + 1.88·3-s + 2.41·5-s − 2.57i·7-s + 0.564·9-s − 0.736·11-s − 4.07i·13-s + 4.55·15-s + (3.99 − 1.02i)17-s + 0.853i·19-s − 4.85i·21-s + 8.20i·23-s + 0.825·25-s − 4.59·27-s − 5.86·29-s + 3.51i·31-s + ⋯
L(s)  = 1  + 1.09·3-s + 1.07·5-s − 0.971i·7-s + 0.188·9-s − 0.221·11-s − 1.13i·13-s + 1.17·15-s + (0.968 − 0.249i)17-s + 0.195i·19-s − 1.05i·21-s + 1.71i·23-s + 0.165·25-s − 0.884·27-s − 1.08·29-s + 0.630i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ 0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27811 - 0.499974i\)
\(L(\frac12)\) \(\approx\) \(2.27811 - 0.499974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.99 + 1.02i)T \)
good3 \( 1 - 1.88T + 3T^{2} \)
5 \( 1 - 2.41T + 5T^{2} \)
7 \( 1 + 2.57iT - 7T^{2} \)
11 \( 1 + 0.736T + 11T^{2} \)
13 \( 1 + 4.07iT - 13T^{2} \)
19 \( 1 - 0.853iT - 19T^{2} \)
23 \( 1 - 8.20iT - 23T^{2} \)
29 \( 1 + 5.86T + 29T^{2} \)
31 \( 1 - 3.51iT - 31T^{2} \)
37 \( 1 - 4.38T + 37T^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 - 1.09iT - 43T^{2} \)
47 \( 1 - 5.42T + 47T^{2} \)
53 \( 1 - 4.66iT - 53T^{2} \)
59 \( 1 + 12.5iT - 59T^{2} \)
61 \( 1 + 8.16T + 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 - 0.511iT - 71T^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 + 1.45iT - 79T^{2} \)
83 \( 1 + 4.35iT - 83T^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 - 11.8iT - 97T^{2} \)
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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

−10.44654348432152348130261169290, −9.772911285064748602352366840105, −9.172160975254084438973572476862, −7.83938751741594569636056147558, −7.58225728531227091146594689091, −6.05171356131925780882500188324, −5.24711920972807677070279965672, −3.68137858030946924966796622807, −2.86442888925216949492305096826, −1.45839693171201570657144845535, 2.01922117259165217700162206635, 2.61216921977835304898133565385, 4.01372817437915929216926512849, 5.46016173819305591901387587636, 6.15378098545041084839498670805, 7.39796220782853952273083951376, 8.490849928435040396211987157421, 9.093300639956331242271943909298, 9.673901690897809974811849250522, 10.68899992892098630320296718728

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