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Properties

Label 2-546-13.3-c1-0-11
Degree $2$
Conductor $546$
Sign $-0.597 + 0.802i$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 2.56·5-s + (0.499 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.28 + 2.21i)10-s + (−0.780 − 1.35i)11-s − 0.999·12-s + (−0.5 − 3.57i)13-s + 0.999·14-s + (−1.28 − 2.21i)15-s + (−0.5 − 0.866i)16-s + (4.06 − 7.03i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 1.14·5-s + (0.204 − 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.405 + 0.701i)10-s + (−0.235 − 0.407i)11-s − 0.288·12-s + (−0.138 − 0.990i)13-s + 0.267·14-s + (−0.330 − 0.572i)15-s + (−0.125 − 0.216i)16-s + (0.985 − 1.70i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.597 + 0.802i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.597 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262767 - 0.523153i\)
\(L(\frac12)\) \(\approx\) \(0.262767 - 0.523153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 3.57i)T \)
good5 \( 1 + 2.56T + 5T^{2} \)
11 \( 1 + (0.780 + 1.35i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.06 + 7.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.780 - 1.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.56 + 6.16i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.06 + 1.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-3.28 - 5.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.62 + 4.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 - 7T + 53T^{2} \)
59 \( 1 + (-1.56 + 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.62 - 4.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.438 + 0.759i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.68 - 11.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.56T + 73T^{2} \)
79 \( 1 + 2.43T + 79T^{2} \)
83 \( 1 - 3.12T + 83T^{2} \)
89 \( 1 + (3.78 + 6.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.56 + 7.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.34836773062637488730080989998, −9.826142664958310189611096007354, −8.649012574907809725469065691998, −8.071875507341444345947254388877, −7.25387520366286420394293603740, −5.64475336053364533715147800330, −4.55000271140865864625408407922, −3.46046007163665308964432715682, −2.68469348378206225240669971916, −0.37352115187422595505984512029, 1.62359781957480983878608191498, 3.53011459871439766322861903628, 4.39369288985163412354208430824, 5.84798205387652453239356846185, 6.81739302926739690352145119055, 7.73248284358295560912386242723, 8.058218068807758135724303121222, 9.202532832951306140696282577089, 10.06411124389661206173449904135, 11.14387192093863938495354857316

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