Properties

Label 2-546-91.45-c1-0-16
Degree $2$
Conductor $546$
Sign $-0.235 + 0.971i$
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 + (1.04 − 3.90i)5-s + (0.258 − 0.965i)6-s + (2.49 − 0.887i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−2.02 − 3.50i)10-s + (−1.31 + 4.89i)11-s + (−0.500 − 0.866i)12-s + (2.15 + 2.88i)13-s + (1.13 − 2.39i)14-s + (−1.04 − 3.90i)15-s − 1.00·16-s − 4.96·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (0.468 − 1.74i)5-s + (0.105 − 0.394i)6-s + (0.942 − 0.335i)7-s + (−0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.639 − 1.10i)10-s + (−0.395 + 1.47i)11-s + (−0.144 − 0.250i)12-s + (0.598 + 0.801i)13-s + (0.303 − 0.638i)14-s + (−0.270 − 1.00i)15-s − 0.250·16-s − 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.235 + 0.971i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53886 - 1.95578i\)
\(L(\frac12)\) \(\approx\) \(1.53886 - 1.95578i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.49 + 0.887i)T \)
13 \( 1 + (-2.15 - 2.88i)T \)
good5 \( 1 + (-1.04 + 3.90i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.31 - 4.89i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + (-1.40 + 0.375i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 5.43iT - 23T^{2} \)
29 \( 1 + (4.24 - 7.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.97 + 1.06i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.552 + 0.552i)T + 37iT^{2} \)
41 \( 1 + (-3.46 + 0.927i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.45 + 4.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.84 + 0.761i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.08 + 3.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.661 - 0.661i)T - 59iT^{2} \)
61 \( 1 + (10.4 + 6.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.88 + 2.38i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.28 + 1.14i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.43 - 12.8i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.32 - 2.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.02 + 1.02i)T + 83iT^{2} \)
89 \( 1 + (-4.52 + 4.52i)T - 89iT^{2} \)
97 \( 1 + (2.62 - 9.79i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.61977137819743456255755299826, −9.376537233296837586922030746161, −9.066486195585868293530769529888, −7.983646328397496271752474896178, −6.99718427043018109650597213031, −5.52957139476080368995562937942, −4.70093611925833884293441636854, −4.09578696117770531599916013831, −2.05269949658594115836837194056, −1.42353381671312997844123225035, 2.46079188988191604721164072520, 3.12410870305296458879308195882, 4.36756304036939667908274306592, 5.81772295030276646428389460390, 6.24209500389731526603515772091, 7.53730884640427702561524351074, 8.193435469799189690406201675538, 9.119354486483943747448794129288, 10.52312303104065919793652115888, 10.90094962139458486671947725918

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