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Properties

Label 2-546-21.5-c1-0-6
Degree $2$
Conductor $546$
Sign $0.694 - 0.719i$
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 + (−1.34 + 1.09i)3-s + (0.499 + 0.866i)4-s + (0.166 − 0.288i)5-s + (1.71 − 0.274i)6-s + (−2.56 + 0.648i)7-s − 0.999i·8-s + (0.611 − 2.93i)9-s + (−0.288 + 0.166i)10-s + (3.91 − 2.26i)11-s + (−1.61 − 0.617i)12-s − i·13-s + (2.54 + 0.720i)14-s + (0.0914 + 0.569i)15-s + (−0.5 + 0.866i)16-s + (2.90 + 5.03i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.775 + 0.630i)3-s + (0.249 + 0.433i)4-s + (0.0744 − 0.128i)5-s + (0.698 − 0.112i)6-s + (−0.969 + 0.245i)7-s − 0.353i·8-s + (0.203 − 0.979i)9-s + (−0.0911 + 0.0526i)10-s + (1.18 − 0.682i)11-s + (−0.467 − 0.178i)12-s − 0.277i·13-s + (0.680 + 0.192i)14-s + (0.0236 + 0.147i)15-s + (−0.125 + 0.216i)16-s + (0.704 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.694 - 0.719i$
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: \((2,\ 546,\ (\ :1/2),\ 0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.674121 + 0.286481i\)
\(L(\frac12)\) \(\approx\) \(0.674121 + 0.286481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 1.09i)T \)
7 \( 1 + (2.56 - 0.648i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-0.166 + 0.288i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.91 + 2.26i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.90 - 5.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.59 + 0.921i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.81 - 1.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.22iT - 29T^{2} \)
31 \( 1 + (8.08 - 4.66i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.22 - 2.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 - 5.15T + 43T^{2} \)
47 \( 1 + (5.51 - 9.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.31 + 3.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.345 - 0.598i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.64 - 5.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.98 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (2.93 - 1.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.99 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.511T + 83T^{2} \)
89 \( 1 + (-8.07 + 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.07iT - 97T^{2} \)
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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

−10.87720083519779681369943744341, −10.06375147793039261626266645885, −9.195457229882419041724210403505, −8.742405551595918001506824259827, −7.20727297052636888849045187166, −6.27193083060086238247183024426, −5.54351385712510934276620792651, −3.97401658036290913884499139381, −3.22724878402016208272994931541, −1.14153847053529213608259486164, 0.70751070851679884603470573595, 2.29945994954877507864646479380, 4.07475112633993569512173332831, 5.42640122817903967293038938948, 6.42997216556257764840306852476, 6.96343808515802445041311343323, 7.71482665233908998523615785544, 9.110183804628943094185550851572, 9.716825861085577917329182530713, 10.62586251854579485619742466536

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