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Properties

Label 2-546-273.173-c1-0-25
Degree $2$
Conductor $546$
Sign $0.964 + 0.263i$
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  + (−0.5 − 0.866i)2-s + (1.57 + 0.717i)3-s + (−0.499 + 0.866i)4-s + (1.41 + 0.815i)5-s + (−0.166 − 1.72i)6-s + (1.06 − 2.42i)7-s + 0.999·8-s + (1.97 + 2.26i)9-s − 1.63i·10-s − 2.07·11-s + (−1.40 + 1.00i)12-s + (3.37 + 1.27i)13-s + (−2.62 + 0.292i)14-s + (1.64 + 2.29i)15-s + (−0.5 − 0.866i)16-s + (1.52 − 2.64i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.910 + 0.414i)3-s + (−0.249 + 0.433i)4-s + (0.631 + 0.364i)5-s + (−0.0680 − 0.703i)6-s + (0.401 − 0.916i)7-s + 0.353·8-s + (0.656 + 0.754i)9-s − 0.515i·10-s − 0.624·11-s + (−0.406 + 0.290i)12-s + (0.935 + 0.352i)13-s + (−0.702 + 0.0782i)14-s + (0.424 + 0.593i)15-s + (−0.125 − 0.216i)16-s + (0.370 − 0.641i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.964 + 0.263i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85486 - 0.248706i\)
\(L(\frac12)\) \(\approx\) \(1.85486 - 0.248706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.57 - 0.717i)T \)
7 \( 1 + (-1.06 + 2.42i)T \)
13 \( 1 + (-3.37 - 1.27i)T \)
good5 \( 1 + (-1.41 - 0.815i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
17 \( 1 + (-1.52 + 2.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (5.29 - 3.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.79 + 1.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.28 - 5.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.87 + 3.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.41 - 4.28i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.69 - 4.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.714 - 0.412i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.25 - 5.34i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.7 + 6.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 15.5iT - 61T^{2} \)
67 \( 1 + 3.27iT - 67T^{2} \)
71 \( 1 + (4.59 + 7.95i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.43 + 2.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.24 - 5.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.85iT - 83T^{2} \)
89 \( 1 + (-0.929 + 0.536i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.58 - 7.94i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.71667310842822440140793420740, −9.767943790508287337833056234938, −9.364648526786610669466921720210, −7.994356264405140339475374620059, −7.68866807340690801569608970756, −6.25097827983872630720248006511, −4.78814692203221390847531256622, −3.80649763017633080931138930446, −2.76067309243278470711563356563, −1.53630976422542505505481555233, 1.47130048048657189577936381867, 2.64540293982442742455914262196, 4.20135174728718692331368256972, 5.67838053610262630689778876046, 6.10589841740148630108570010922, 7.58685921617029060617340481159, 8.182388881621555664435184058001, 8.900904902707216390105983517110, 9.636534657265676382822705732146, 10.54321146345210000759930423725

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