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Properties

Label 2-546-273.257-c1-0-24
Degree $2$
Conductor $546$
Sign $0.758 + 0.651i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  − 2-s + (0.841 − 1.51i)3-s + 4-s + (2.84 + 1.64i)5-s + (−0.841 + 1.51i)6-s + (1.87 − 1.86i)7-s − 8-s + (−1.58 − 2.54i)9-s + (−2.84 − 1.64i)10-s + (−1.03 + 1.79i)11-s + (0.841 − 1.51i)12-s + (3.53 − 0.716i)13-s + (−1.87 + 1.86i)14-s + (4.88 − 2.92i)15-s + 16-s + 1.12·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.486 − 0.873i)3-s + 0.5·4-s + (1.27 + 0.735i)5-s + (−0.343 + 0.617i)6-s + (0.707 − 0.706i)7-s − 0.353·8-s + (−0.527 − 0.849i)9-s + (−0.900 − 0.519i)10-s + (−0.312 + 0.540i)11-s + (0.243 − 0.436i)12-s + (0.980 − 0.198i)13-s + (−0.500 + 0.499i)14-s + (1.26 − 0.755i)15-s + 0.250·16-s + 0.272·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52344 - 0.564393i\)
\(L(\frac12)\) \(\approx\) \(1.52344 - 0.564393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.841 + 1.51i)T \)
7 \( 1 + (-1.87 + 1.86i)T \)
13 \( 1 + (-3.53 + 0.716i)T \)
good5 \( 1 + (-2.84 - 1.64i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.03 - 1.79i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.12T + 17T^{2} \)
19 \( 1 + (0.505 + 0.875i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.50iT - 23T^{2} \)
29 \( 1 + (7.97 - 4.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.86 + 3.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.97iT - 37T^{2} \)
41 \( 1 + (-8.52 + 4.92i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.35 + 5.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.05 - 0.609i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.05 - 2.92i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.80iT - 59T^{2} \)
61 \( 1 + (0.209 - 0.120i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.2 + 5.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.94 + 6.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.878 - 1.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.48 - 4.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.999iT - 83T^{2} \)
89 \( 1 - 7.61iT - 89T^{2} \)
97 \( 1 + (3.21 - 5.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.72367908123948205155921363488, −9.707663165003313077986930454904, −9.004213145835633946057513491606, −7.86156167523575245476568945968, −7.28484627733128831736794270694, −6.37475438312587610137841645753, −5.49383632533697184859611057847, −3.55569974770693618326369678968, −2.24791726365631091470158124774, −1.38900311083116223027143907340, 1.62595666343983434029003753088, 2.70280248648935893099008695580, 4.28407894145372736406602427025, 5.57903050683342800277719898818, 5.92908453610225991079813327526, 7.76816956493017183970701554774, 8.609880453995901728179997492304, 9.094182540657047972920867012821, 9.771514934297169301794168985410, 10.78076048630651407896120794716

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