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Properties

Label 2-546-13.4-c1-0-5
Degree $2$
Conductor $546$
Sign $0.667 + 0.744i$
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  + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + 0.332i·5-s + (−0.866 + 0.499i)6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.166 − 0.288i)10-s + (2.26 + 1.30i)11-s + 0.999·12-s + (3.41 + 1.16i)13-s + 0.999·14-s + (0.288 + 0.166i)15-s + (−0.5 + 0.866i)16-s + (−1.94 − 3.36i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 0.148i·5-s + (−0.353 + 0.204i)6-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.0526 − 0.0911i)10-s + (0.681 + 0.393i)11-s + 0.288·12-s + (0.946 + 0.323i)13-s + 0.267·14-s + (0.0744 + 0.0429i)15-s + (−0.125 + 0.216i)16-s + (−0.471 − 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13165 - 0.505660i\)
\(L(\frac12)\) \(\approx\) \(1.13165 - 0.505660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.41 - 1.16i)T \)
good5 \( 1 - 0.332iT - 5T^{2} \)
11 \( 1 + (-2.26 - 1.30i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.94 + 3.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.85 + 2.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.10 + 3.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.593 + 1.02i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.07iT - 31T^{2} \)
37 \( 1 + (-0.499 - 0.288i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.451 - 0.260i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.53 - 2.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.0iT - 47T^{2} \)
53 \( 1 + 9.71T + 53T^{2} \)
59 \( 1 + (-9.07 + 5.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.71 + 6.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 5.96i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.818 - 0.472i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.66iT - 73T^{2} \)
79 \( 1 + 0.943T + 79T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + (-2.92 - 1.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.03 - 2.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.75164477818556914080396249984, −9.567062965184724702666482549688, −9.015570408264289140388369077208, −8.207444072067814404740625425372, −6.88884640513958376694074075719, −6.67223679115452552462770118025, −5.02033403673555551678467121960, −3.57077956221773455200562312936, −2.54695127516940982732816701043, −1.09057123630961569093770172167, 1.27720965018775831276361634425, 3.14060414376862508795376117042, 4.15570252696066319005374244641, 5.56977303727984322204221486427, 6.34658706230233626192044419964, 7.50268761316838694964059406810, 8.400953944253608679913008887350, 9.144509710331623613908121675418, 9.839699141875592161504024291142, 10.83355609530452157101228701212

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