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Properties

Label 2-546-273.269-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.218 + 0.975i$
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  + i·2-s + (0.0345 + 1.73i)3-s − 4-s + (−0.699 − 1.21i)5-s + (−1.73 + 0.0345i)6-s + (−2.51 + 0.831i)7-s − i·8-s + (−2.99 + 0.119i)9-s + (1.21 − 0.699i)10-s + (−1.09 + 0.632i)11-s + (−0.0345 − 1.73i)12-s + (3.58 − 0.379i)13-s + (−0.831 − 2.51i)14-s + (2.07 − 1.25i)15-s + 16-s − 6.86·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.0199 + 0.999i)3-s − 0.5·4-s + (−0.312 − 0.541i)5-s + (−0.706 + 0.0141i)6-s + (−0.949 + 0.314i)7-s − 0.353i·8-s + (−0.999 + 0.0399i)9-s + (0.383 − 0.221i)10-s + (−0.330 + 0.190i)11-s + (−0.00998 − 0.499i)12-s + (0.994 − 0.105i)13-s + (−0.222 − 0.671i)14-s + (0.535 − 0.323i)15-s + 0.250·16-s − 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0622594 - 0.0777093i\)
\(L(\frac12)\) \(\approx\) \(0.0622594 - 0.0777093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.0345 - 1.73i)T \)
7 \( 1 + (2.51 - 0.831i)T \)
13 \( 1 + (-3.58 + 0.379i)T \)
good5 \( 1 + (0.699 + 1.21i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.09 - 0.632i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.86T + 17T^{2} \)
19 \( 1 + (3.99 + 2.30i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.51iT - 23T^{2} \)
29 \( 1 + (4.55 + 2.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.41 - 2.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.91T + 37T^{2} \)
41 \( 1 + (3.22 - 5.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.08 - 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.58 - 2.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.444 - 0.256i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + (2.66 + 1.53i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.312 - 0.540i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.78 - 2.18i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.24 + 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.13 - 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (-2.18 + 1.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.18106752092375782301197626692, −10.48322467133235946573031063995, −9.401252609263035269877198538044, −8.802251711865637785993700932798, −8.187710153508664843136194788472, −6.63798274440501371077793562380, −6.03195354236467162868691671159, −4.75467957163529273291417329676, −4.14423896243173865535959400834, −2.80923672067119248417490895681, 0.05436736173318556236401667625, 1.88625348530871941099977771594, 3.10467912750364178177227409156, 3.99938679655648355664994616325, 5.71028315457352925859513903967, 6.61575337467965572636882168807, 7.34696590246498227749030837087, 8.562704067638758194339966422374, 9.144880776141465624881034366292, 10.65684607859786393243670418758

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