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Properties

Label 2-546-39.5-c1-0-9
Degree $2$
Conductor $546$
Sign $0.620 - 0.784i$
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 + (−1.18 + 1.26i)3-s + 1.00i·4-s + (2.80 + 2.80i)5-s + (1.73 − 0.0557i)6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.192 − 2.99i)9-s − 3.96i·10-s + (3.20 − 3.20i)11-s + (−1.26 − 1.18i)12-s + (3.52 + 0.750i)13-s − 1.00i·14-s + (−6.85 + 0.220i)15-s − 1.00·16-s + 3.46·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.684 + 0.729i)3-s + 0.500i·4-s + (1.25 + 1.25i)5-s + (0.706 − 0.0227i)6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.0642 − 0.997i)9-s − 1.25i·10-s + (0.967 − 0.967i)11-s + (−0.364 − 0.342i)12-s + (0.978 + 0.208i)13-s − 0.267i·14-s + (−1.77 + 0.0569i)15-s − 0.250·16-s + 0.839·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.620 - 0.784i$
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: \((2,\ 546,\ (\ :1/2),\ 0.620 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11426 + 0.539523i\)
\(L(\frac12)\) \(\approx\) \(1.11426 + 0.539523i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.26i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-3.52 - 0.750i)T \)
good5 \( 1 + (-2.80 - 2.80i)T + 5iT^{2} \)
11 \( 1 + (-3.20 + 3.20i)T - 11iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (1.81 - 1.81i)T - 19iT^{2} \)
23 \( 1 + 0.728T + 23T^{2} \)
29 \( 1 + 6.96iT - 29T^{2} \)
31 \( 1 + (7.26 - 7.26i)T - 31iT^{2} \)
37 \( 1 + (1.56 + 1.56i)T + 37iT^{2} \)
41 \( 1 + (-3.59 - 3.59i)T + 41iT^{2} \)
43 \( 1 + 7.44iT - 43T^{2} \)
47 \( 1 + (-7.17 + 7.17i)T - 47iT^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 + (7.33 - 7.33i)T - 59iT^{2} \)
61 \( 1 + 0.300T + 61T^{2} \)
67 \( 1 + (3.02 - 3.02i)T - 67iT^{2} \)
71 \( 1 + (3.36 + 3.36i)T + 71iT^{2} \)
73 \( 1 + (6.89 + 6.89i)T + 73iT^{2} \)
79 \( 1 + 5.87T + 79T^{2} \)
83 \( 1 + (7.55 + 7.55i)T + 83iT^{2} \)
89 \( 1 + (-0.819 + 0.819i)T - 89iT^{2} \)
97 \( 1 + (-1.37 + 1.37i)T - 97iT^{2} \)
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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

−10.66325593563974292248538521633, −10.36656566151076902939729603795, −9.295170071194478522404215092984, −8.756077261876263538508636924708, −7.17629772596076128795255759424, −6.05304994439241813612664225193, −5.78143076003272727886504421445, −3.95913097465511382841766629629, −3.07178771444124084399454649159, −1.52546437375092169635505502676, 1.11973554140540108762840973935, 1.84014862623174512449303415469, 4.41903842207405057299269388477, 5.41251141683379532170330190159, 6.05893632561941313789786211812, 6.97466577728303495849243891212, 7.990914798160070243284826428486, 8.940608544425035467769909265203, 9.610156733762511827045336269729, 10.56596438968368185453285694685

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