Properties

Label 2-546-273.101-c1-0-19
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: $\chi_{546} (101, \cdot )$
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

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

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