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Properties

Label 2-546-273.257-c1-0-14
Degree $2$
Conductor $546$
Sign $0.254 - 0.967i$
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 + (−0.942 + 1.45i)3-s + 4-s + (3.27 + 1.89i)5-s + (−0.942 + 1.45i)6-s + (0.475 + 2.60i)7-s + 8-s + (−1.22 − 2.73i)9-s + (3.27 + 1.89i)10-s + (2.90 − 5.03i)11-s + (−0.942 + 1.45i)12-s + (−0.879 + 3.49i)13-s + (0.475 + 2.60i)14-s + (−5.84 + 2.98i)15-s + 16-s − 4.53·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.544 + 0.838i)3-s + 0.5·4-s + (1.46 + 0.846i)5-s + (−0.384 + 0.593i)6-s + (0.179 + 0.983i)7-s + 0.353·8-s + (−0.407 − 0.913i)9-s + (1.03 + 0.598i)10-s + (0.875 − 1.51i)11-s + (−0.272 + 0.419i)12-s + (−0.243 + 0.969i)13-s + (0.126 + 0.695i)14-s + (−1.50 + 0.769i)15-s + 0.250·16-s − 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.254 - 0.967i$
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: \((2,\ 546,\ (\ :1/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86260 + 1.43606i\)
\(L(\frac12)\) \(\approx\) \(1.86260 + 1.43606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.942 - 1.45i)T \)
7 \( 1 + (-0.475 - 2.60i)T \)
13 \( 1 + (0.879 - 3.49i)T \)
good5 \( 1 + (-3.27 - 1.89i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.90 + 5.03i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + (1.42 + 2.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.84iT - 23T^{2} \)
29 \( 1 + (2.20 - 1.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.824 + 1.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.999iT - 37T^{2} \)
41 \( 1 + (3.48 - 2.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.445 - 0.772i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.23 - 3.59i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.41 + 3.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + (3.15 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.24 + 5.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.77 + 11.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.19 - 3.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.40 - 14.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 - 2.18iT - 89T^{2} \)
97 \( 1 + (-6.30 + 10.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.20873254241975377688305439501, −10.29190747212401147784609155232, −9.187631623200363460449529855334, −8.825465670668406784145793589186, −6.59461669045162107263982344145, −6.33969404483974176576509621760, −5.52991040589714187112338749480, −4.49449051217128183619526709712, −3.16770751426742302269165691838, −2.12498761741351749767149080240, 1.36086594889187205671110061550, 2.17289717693867389361867282089, 4.21130040301716887562873553793, 5.11758637593950086210920839944, 5.90990503076725300945065821678, 6.86052174039741547268461226462, 7.54498027996637395381403742257, 8.885772249999982963208081802614, 9.986577024161709329760961138067, 10.57960376846633554645015064448

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