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Properties

Label 2-546-273.257-c1-0-4
Degree $2$
Conductor $546$
Sign $-0.0821 - 0.996i$
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  − 2-s + (−1.31 − 1.12i)3-s + 4-s + (1.62 + 0.936i)5-s + (1.31 + 1.12i)6-s + (2.04 + 1.68i)7-s − 8-s + (0.453 + 2.96i)9-s + (−1.62 − 0.936i)10-s + (−2.54 + 4.41i)11-s + (−1.31 − 1.12i)12-s + (−3.20 − 1.64i)13-s + (−2.04 − 1.68i)14-s + (−1.07 − 3.06i)15-s + 16-s − 2.96·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.758 − 0.651i)3-s + 0.5·4-s + (0.725 + 0.418i)5-s + (0.536 + 0.460i)6-s + (0.772 + 0.635i)7-s − 0.353·8-s + (0.151 + 0.988i)9-s + (−0.512 − 0.296i)10-s + (−0.768 + 1.33i)11-s + (−0.379 − 0.325i)12-s + (−0.889 − 0.456i)13-s + (−0.545 − 0.449i)14-s + (−0.277 − 0.790i)15-s + 0.250·16-s − 0.718·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0821 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0821 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0821 - 0.996i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0821 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426879 + 0.463508i\)
\(L(\frac12)\) \(\approx\) \(0.426879 + 0.463508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.31 + 1.12i)T \)
7 \( 1 + (-2.04 - 1.68i)T \)
13 \( 1 + (3.20 + 1.64i)T \)
good5 \( 1 + (-1.62 - 0.936i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.54 - 4.41i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.96T + 17T^{2} \)
19 \( 1 + (1.30 + 2.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.06iT - 23T^{2} \)
29 \( 1 + (7.30 - 4.21i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.09 - 7.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.37iT - 37T^{2} \)
41 \( 1 + (4.85 - 2.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.87 - 8.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.70 - 1.56i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.21 + 4.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.20iT - 59T^{2} \)
61 \( 1 + (-5.11 + 2.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.56 - 5.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.27 + 12.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.99 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.14 - 8.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.63iT - 83T^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 + (-5.83 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.95774683088834353700743014615, −10.20315358165206244458354250045, −9.436566694965487571415010626969, −8.214546784539227052581742391430, −7.37645343875713699136715619433, −6.72477270101133656512552815373, −5.49882352769151024367416635167, −4.90881799804957751580752428439, −2.47249660159860802483054488157, −1.81107597650162853993018561677, 0.47544378790942269380736355827, 2.18010573368322773840347830973, 3.97477345839112847059205901707, 5.08878203224662537006070069571, 5.88051877755639077232625346907, 6.90424633895079666507834013447, 8.111899261654925232292107091517, 8.880072446425139166250268205078, 9.897900299971038275009537396691, 10.43646787982456647644998203096

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