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Properties

Label 2-546-273.242-c1-0-31
Degree $2$
Conductor $546$
Sign $0.660 + 0.750i$
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  + (0.965 + 0.258i)2-s + (1.64 − 0.537i)3-s + (0.866 + 0.499i)4-s + (−3.24 − 0.868i)5-s + (1.72 − 0.0931i)6-s + (−1.25 − 2.32i)7-s + (0.707 + 0.707i)8-s + (2.42 − 1.77i)9-s + (−2.90 − 1.67i)10-s + (6.11 − 1.63i)11-s + (1.69 + 0.357i)12-s + (−0.0266 − 3.60i)13-s + (−0.608 − 2.57i)14-s + (−5.80 + 0.312i)15-s + (0.500 + 0.866i)16-s + (−2.03 + 3.52i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.950 − 0.310i)3-s + (0.433 + 0.249i)4-s + (−1.44 − 0.388i)5-s + (0.706 − 0.0380i)6-s + (−0.474 − 0.880i)7-s + (0.249 + 0.249i)8-s + (0.807 − 0.590i)9-s + (−0.918 − 0.530i)10-s + (1.84 − 0.493i)11-s + (0.489 + 0.103i)12-s + (−0.00740 − 0.999i)13-s + (−0.162 − 0.688i)14-s + (−1.49 + 0.0806i)15-s + (0.125 + 0.216i)16-s + (−0.493 + 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.660 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17899 - 0.985383i\)
\(L(\frac12)\) \(\approx\) \(2.17899 - 0.985383i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-1.64 + 0.537i)T \)
7 \( 1 + (1.25 + 2.32i)T \)
13 \( 1 + (0.0266 + 3.60i)T \)
good5 \( 1 + (3.24 + 0.868i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-6.11 + 1.63i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.03 - 3.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.600 + 2.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.74 - 4.74i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.127iT - 29T^{2} \)
31 \( 1 + (-1.65 + 0.443i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.58 + 2.03i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.12 - 7.12i)T - 41iT^{2} \)
43 \( 1 - 2.56iT - 43T^{2} \)
47 \( 1 + (2.54 - 9.48i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.68 + 1.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.18 + 1.65i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.79 - 3.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.15 + 0.845i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.71 + 8.71i)T - 71iT^{2} \)
73 \( 1 + (-1.09 - 4.08i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.01 - 5.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.395 + 0.395i)T - 83iT^{2} \)
89 \( 1 + (-0.836 + 3.12i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.43 - 2.43i)T + 97iT^{2} \)
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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

−10.95322975708206542236760184998, −9.634615761238224877596280371003, −8.642659996755249857591045277772, −7.945215424007876806435366056640, −7.08948618503378357865407771338, −6.39258649525168078542875772304, −4.63348322031902466024256191197, −3.67384378584312032263396066512, −3.37975873010696988963098481447, −1.14765253211792763649933742131, 2.07226356332421593679782191632, 3.38213907664124446225162610733, 3.96618929535482752024364520561, 4.89325226569239221664318120673, 6.84516204398526255108017586662, 6.90256069387764873654270311410, 8.462108410795588936889343441304, 9.040670633548785247745632927273, 9.972041490187572245619376034243, 11.19349988575598335030457608012

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