Properties

Label 2-546-91.59-c1-0-13
Degree $2$
Conductor $546$
Sign $0.0651 + 0.997i$
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  + (0.707 + 0.707i)2-s + (−0.866 + 0.5i)3-s + 1.00i·4-s + (−2.31 − 0.620i)5-s + (−0.965 − 0.258i)6-s + (−2.57 − 0.617i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.19 − 2.07i)10-s + (3.24 + 0.869i)11-s + (−0.500 − 0.866i)12-s + (2.14 − 2.89i)13-s + (−1.38 − 2.25i)14-s + (2.31 − 0.620i)15-s − 1.00·16-s − 6.37·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.499 + 0.288i)3-s + 0.500i·4-s + (−1.03 − 0.277i)5-s + (−0.394 − 0.105i)6-s + (−0.972 − 0.233i)7-s + (−0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.378 − 0.656i)10-s + (0.978 + 0.262i)11-s + (−0.144 − 0.250i)12-s + (0.595 − 0.803i)13-s + (−0.369 − 0.602i)14-s + (0.597 − 0.160i)15-s − 0.250·16-s − 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0651 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0651 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0651 + 0.997i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0651 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.390601 - 0.365935i\)
\(L(\frac12)\) \(\approx\) \(0.390601 - 0.365935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.57 + 0.617i)T \)
13 \( 1 + (-2.14 + 2.89i)T \)
good5 \( 1 + (2.31 + 0.620i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.24 - 0.869i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 6.37T + 17T^{2} \)
19 \( 1 + (1.65 + 6.16i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.51iT - 23T^{2} \)
29 \( 1 + (-1.87 + 3.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.56 + 5.84i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.23 - 2.23i)T - 37iT^{2} \)
41 \( 1 + (-2.36 - 8.82i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.35 - 4.24i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.14 - 8.00i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.65 - 8.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.73 + 1.73i)T + 59iT^{2} \)
61 \( 1 + (6.53 + 3.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.93 + 7.21i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.165 + 0.616i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.6 + 2.86i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.75 + 4.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.287 + 0.287i)T - 83iT^{2} \)
89 \( 1 + (1.95 + 1.95i)T + 89iT^{2} \)
97 \( 1 + (3.55 + 0.953i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.90341172705958467906601746812, −9.590918711977561013078683338543, −8.764916175381325044182309801640, −7.80563948538582819569379561510, −6.51612784273916217584981739394, −6.37453380536623092654065503694, −4.60508065801419236438485684807, −4.22824878733803787182378515826, −2.99237776723044038674869330863, −0.27600054826262067180383062407, 1.76428642893850659953751986396, 3.56933266460586043896921382724, 3.96444050102616271726276872283, 5.47027635096123546506836737761, 6.56388920090476087410670778546, 6.98403284247337848724136917882, 8.492731283937491606698183411612, 9.288822636600841361194328223259, 10.44764026269252981787082240727, 11.24885401182805910574275810747

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