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Properties

Label 2-546-91.4-c1-0-13
Degree $2$
Conductor $546$
Sign $0.662 + 0.749i$
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  − i·2-s + (0.5 + 0.866i)3-s − 4-s + (0.813 − 0.469i)5-s + (0.866 − 0.5i)6-s + (2.42 − 1.06i)7-s + i·8-s + (−0.499 + 0.866i)9-s + (−0.469 − 0.813i)10-s + (1.02 − 0.592i)11-s + (−0.5 − 0.866i)12-s + (−2.78 − 2.28i)13-s + (−1.06 − 2.42i)14-s + (0.813 + 0.469i)15-s + 16-s + 4.44·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 + 0.499i)3-s − 0.5·4-s + (0.363 − 0.210i)5-s + (0.353 − 0.204i)6-s + (0.915 − 0.402i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.148 − 0.257i)10-s + (0.309 − 0.178i)11-s + (−0.144 − 0.249i)12-s + (−0.772 − 0.634i)13-s + (−0.284 − 0.647i)14-s + (0.210 + 0.121i)15-s + 0.250·16-s + 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61295 - 0.726976i\)
\(L(\frac12)\) \(\approx\) \(1.61295 - 0.726976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.42 + 1.06i)T \)
13 \( 1 + (2.78 + 2.28i)T \)
good5 \( 1 + (-0.813 + 0.469i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.02 + 0.592i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + (-2.19 - 1.26i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 + (-4.86 + 8.41i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.13 - 0.656i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.99iT - 37T^{2} \)
41 \( 1 + (3.93 + 2.27i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.49 + 6.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.13 + 2.96i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.41 + 4.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.56iT - 59T^{2} \)
61 \( 1 + (6.79 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.0 - 6.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.59 - 3.80i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.28 - 4.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.88 + 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 + (11.3 - 6.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.32816265377722861315553139226, −10.14903481575564989657311435971, −9.068028538477221172803340321424, −8.173910811508805100010347453845, −7.37182011731021728483766650700, −5.66645027303017340217509983583, −4.92962146135346057485998126606, −3.86778896490085176865597712533, −2.71023934173279386951431837579, −1.25131761854073082846487029069, 1.53226803201635069116792718134, 2.94917551639980725287890159655, 4.55626652881492371195514551418, 5.42419332364981768925528807933, 6.50247924230873438379195284261, 7.34410175944933951420670395657, 8.097963270038844494305787222022, 9.035810474171602936933561494493, 9.763729978115947101659187866432, 10.91483298284831067535795078822

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