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Properties

Label 2-546-21.5-c1-0-9
Degree $2$
Conductor $546$
Sign $-0.703 - 0.710i$
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 + (−0.0921 + 1.72i)3-s + (0.499 + 0.866i)4-s + (0.890 − 1.54i)5-s + (−0.944 + 1.45i)6-s + (−1.51 + 2.16i)7-s + 0.999i·8-s + (−2.98 − 0.318i)9-s + (1.54 − 0.890i)10-s + (−3.61 + 2.08i)11-s + (−1.54 + 0.785i)12-s + i·13-s + (−2.39 + 1.11i)14-s + (2.58 + 1.68i)15-s + (−0.5 + 0.866i)16-s + (3.81 + 6.60i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.0531 + 0.998i)3-s + (0.249 + 0.433i)4-s + (0.398 − 0.689i)5-s + (−0.385 + 0.592i)6-s + (−0.573 + 0.819i)7-s + 0.353i·8-s + (−0.994 − 0.106i)9-s + (0.487 − 0.281i)10-s + (−1.09 + 0.629i)11-s + (−0.445 + 0.226i)12-s + 0.277i·13-s + (−0.640 + 0.298i)14-s + (0.667 + 0.434i)15-s + (−0.125 + 0.216i)16-s + (0.924 + 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.703 - 0.710i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661623 + 1.58541i\)
\(L(\frac12)\) \(\approx\) \(0.661623 + 1.58541i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0921 - 1.72i)T \)
7 \( 1 + (1.51 - 2.16i)T \)
13 \( 1 - iT \)
good5 \( 1 + (-0.890 + 1.54i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.61 - 2.08i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.81 - 6.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.62 + 1.51i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.70 - 3.87i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.65iT - 29T^{2} \)
31 \( 1 + (1.78 - 1.03i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.95 + 6.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.69T + 41T^{2} \)
43 \( 1 - 0.322T + 43T^{2} \)
47 \( 1 + (2.04 - 3.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.2 + 5.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.86 - 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.38 + 4.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.25 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.91iT - 71T^{2} \)
73 \( 1 + (-1.14 + 0.661i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.64 + 9.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.38T + 83T^{2} \)
89 \( 1 + (1.23 - 2.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.52iT - 97T^{2} \)
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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

−11.08218613731217593358413399227, −10.19236323926834335501476916781, −9.328193437599504863466728744594, −8.644318884856253402463292664739, −7.58774984337014140198240581835, −6.09613847680943422033366998740, −5.51664593365262308587270228839, −4.70865057321478240659280600554, −3.57473245463248585444803421289, −2.35701529678581234711487347399, 0.814783873284100228910843566226, 2.69207509527722744474333937641, 3.15804386345413678058729221984, 4.96420188606280740726375954568, 5.89807896856475815220449662443, 6.86328280375559599000584956861, 7.39919512566371307057422297235, 8.598281395943667697369193290917, 9.933560215977703883381154946550, 10.66980341621518574325675768474

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