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Properties

Label 2-546-39.5-c1-0-18
Degree $2$
Conductor $546$
Sign $0.0338 + 0.999i$
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 + (−0.337 − 1.69i)3-s + 1.00i·4-s + (2.19 + 2.19i)5-s + (−0.962 + 1.44i)6-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−2.77 + 1.14i)9-s − 3.09i·10-s + (1.27 − 1.27i)11-s + (1.69 − 0.337i)12-s + (−3.23 − 1.58i)13-s + 1.00i·14-s + (2.98 − 4.46i)15-s − 1.00·16-s + 3.68·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.194 − 0.980i)3-s + 0.500i·4-s + (0.980 + 0.980i)5-s + (−0.392 + 0.587i)6-s + (−0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.923 + 0.382i)9-s − 0.980i·10-s + (0.384 − 0.384i)11-s + (0.490 − 0.0974i)12-s + (−0.897 − 0.440i)13-s + 0.267i·14-s + (0.770 − 1.15i)15-s − 0.250·16-s + 0.892·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0338 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0338 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0338 + 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0338 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850076 - 0.821753i\)
\(L(\frac12)\) \(\approx\) \(0.850076 - 0.821753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.337 + 1.69i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (3.23 + 1.58i)T \)
good5 \( 1 + (-2.19 - 2.19i)T + 5iT^{2} \)
11 \( 1 + (-1.27 + 1.27i)T - 11iT^{2} \)
17 \( 1 - 3.68T + 17T^{2} \)
19 \( 1 + (-5.26 + 5.26i)T - 19iT^{2} \)
23 \( 1 - 8.15T + 23T^{2} \)
29 \( 1 + 9.50iT - 29T^{2} \)
31 \( 1 + (0.125 - 0.125i)T - 31iT^{2} \)
37 \( 1 + (0.328 + 0.328i)T + 37iT^{2} \)
41 \( 1 + (2.21 + 2.21i)T + 41iT^{2} \)
43 \( 1 + 1.43iT - 43T^{2} \)
47 \( 1 + (0.805 - 0.805i)T - 47iT^{2} \)
53 \( 1 - 5.59iT - 53T^{2} \)
59 \( 1 + (-1.49 + 1.49i)T - 59iT^{2} \)
61 \( 1 + 8.14T + 61T^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 67iT^{2} \)
71 \( 1 + (0.752 + 0.752i)T + 71iT^{2} \)
73 \( 1 + (-3.59 - 3.59i)T + 73iT^{2} \)
79 \( 1 - 4.38T + 79T^{2} \)
83 \( 1 + (-9.35 - 9.35i)T + 83iT^{2} \)
89 \( 1 + (-3.19 + 3.19i)T - 89iT^{2} \)
97 \( 1 + (-1.79 + 1.79i)T - 97iT^{2} \)
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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

−10.60335413655622057753937805616, −9.780910114104967531174998174308, −9.021740661239789027744391760445, −7.65643092614784873538321726700, −7.11387244123781051244734013320, −6.21854439921630229321193843368, −5.19237432279635881490334040548, −3.13314125143481705650662543856, −2.50048578588406985200145409669, −0.946536480488979162954107322543, 1.42840454817281527534580902899, 3.25970535848757692973950401620, 4.94217562170613891296059503260, 5.25405456507213699908457932198, 6.29596311338459615350585462504, 7.48088837608944936845350959856, 8.756582089209074405713617885384, 9.372837417102640915340964561659, 9.776136506460649447385515815498, 10.63381288856660900525336888291

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