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Properties

Label 2-546-273.257-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.363 - 0.931i$
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 + (0.403 + 1.68i)3-s + 4-s + (−1.80 − 1.04i)5-s + (0.403 + 1.68i)6-s + (−0.800 + 2.52i)7-s + 8-s + (−2.67 + 1.35i)9-s + (−1.80 − 1.04i)10-s + (−1.07 + 1.86i)11-s + (0.403 + 1.68i)12-s + (0.217 + 3.59i)13-s + (−0.800 + 2.52i)14-s + (1.02 − 3.45i)15-s + 16-s + 0.557·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.232 + 0.972i)3-s + 0.5·4-s + (−0.806 − 0.465i)5-s + (0.164 + 0.687i)6-s + (−0.302 + 0.953i)7-s + 0.353·8-s + (−0.891 + 0.452i)9-s + (−0.570 − 0.329i)10-s + (−0.325 + 0.563i)11-s + (0.116 + 0.486i)12-s + (0.0602 + 0.998i)13-s + (−0.213 + 0.673i)14-s + (0.264 − 0.892i)15-s + 0.250·16-s + 0.135·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.363 - 0.931i$
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: \((2,\ 546,\ (\ :1/2),\ -0.363 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999641 + 1.46295i\)
\(L(\frac12)\) \(\approx\) \(0.999641 + 1.46295i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.403 - 1.68i)T \)
7 \( 1 + (0.800 - 2.52i)T \)
13 \( 1 + (-0.217 - 3.59i)T \)
good5 \( 1 + (1.80 + 1.04i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.07 - 1.86i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.557T + 17T^{2} \)
19 \( 1 + (-1.94 - 3.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.07iT - 23T^{2} \)
29 \( 1 + (-6.10 + 3.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.21 - 5.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.31iT - 37T^{2} \)
41 \( 1 + (-0.532 + 0.307i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.507 - 0.292i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.68 - 3.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.56iT - 59T^{2} \)
61 \( 1 + (7.10 - 4.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.3 - 7.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.52 + 11.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.198 - 0.344i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.73 + 9.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 - 8.70iT - 89T^{2} \)
97 \( 1 + (-1.64 + 2.85i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.22536072823333383581796205077, −10.19442732994135212371785013602, −9.324406354859250467895768715811, −8.497300561358890889307964434566, −7.58809608306316572900802857716, −6.23522554685172561725064627240, −5.23799470702911051499159612884, −4.41878067426002771572710130971, −3.55220625453447954156892757416, −2.32453977147502063258826147385, 0.802745240387891400876195489918, 2.85408826733103076549138653957, 3.44199790356811513541871081964, 4.81760098554538584362504778788, 6.13830908900392469875640456507, 6.88496552347244603045829530593, 7.74378788045663015581624149269, 8.255882027470197735508313720404, 9.810110924532402722902021750277, 10.92894873840283266041071458842

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