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Properties

Label 2-546-273.44-c1-0-12
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: $\chi_{546} (317, \cdot )$
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

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

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