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Properties

Label 2-546-273.17-c1-0-20
Degree $2$
Conductor $546$
Sign $0.153 + 0.988i$
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 + (0.787 + 1.54i)3-s + 4-s + (−3.27 + 1.89i)5-s + (−0.787 − 1.54i)6-s + (0.475 − 2.60i)7-s − 8-s + (−1.76 + 2.42i)9-s + (3.27 − 1.89i)10-s + (−2.90 − 5.03i)11-s + (0.787 + 1.54i)12-s + (−0.879 − 3.49i)13-s + (−0.475 + 2.60i)14-s + (−5.50 − 3.56i)15-s + 16-s + 4.53·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.454 + 0.890i)3-s + 0.5·4-s + (−1.46 + 0.846i)5-s + (−0.321 − 0.629i)6-s + (0.179 − 0.983i)7-s − 0.353·8-s + (−0.586 + 0.809i)9-s + (1.03 − 0.598i)10-s + (−0.875 − 1.51i)11-s + (0.227 + 0.445i)12-s + (−0.243 − 0.969i)13-s + (−0.126 + 0.695i)14-s + (−1.42 − 0.921i)15-s + 0.250·16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.153 + 0.988i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.330770 - 0.283484i\)
\(L(\frac12)\) \(\approx\) \(0.330770 - 0.283484i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.787 - 1.54i)T \)
7 \( 1 + (-0.475 + 2.60i)T \)
13 \( 1 + (0.879 + 3.49i)T \)
good5 \( 1 + (3.27 - 1.89i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.90 + 5.03i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.53T + 17T^{2} \)
19 \( 1 + (1.42 - 2.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.84iT - 23T^{2} \)
29 \( 1 + (-2.20 - 1.27i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.824 - 1.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.999iT - 37T^{2} \)
41 \( 1 + (-3.48 - 2.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.445 + 0.772i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.23 - 3.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.41 + 3.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + (3.15 + 1.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.24 - 5.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.77 + 11.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.19 + 3.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.40 + 14.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 - 2.18iT - 89T^{2} \)
97 \( 1 + (-6.30 - 10.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.70373636181347870408696068473, −10.03667889892999888396914866483, −8.554754660240830908813349575857, −7.924763197749805692392028392731, −7.58591891049918860880816567599, −6.12196684898829301744057075236, −4.73287913135303585237060661596, −3.40420909041097684336417896442, −3.09165245338908250182386174181, −0.30048853884473995321285799953, 1.57149575199092530779589812068, 2.80395744338280181429019219841, 4.32794280704198810964482470757, 5.47603932548345764969483431964, 6.98262324486034918831135247384, 7.63216255781217130437509831424, 8.216913716520022804927843668173, 9.045727193485396958802629746494, 9.765567801185357652064079282516, 11.31767670649421141258756141538

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