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Properties

Label 2-546-273.194-c1-0-25
Degree $2$
Conductor $546$
Sign $0.890 + 0.455i$
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 + (1.71 + 0.230i)3-s + (−0.499 − 0.866i)4-s + (−1.00 − 0.578i)5-s + (−1.05 + 1.37i)6-s + (0.296 − 2.62i)7-s + 0.999·8-s + (2.89 + 0.790i)9-s + (1.00 − 0.578i)10-s + (−1.85 − 3.21i)11-s + (−0.658 − 1.60i)12-s + (0.361 − 3.58i)13-s + (2.12 + 1.57i)14-s + (−1.58 − 1.22i)15-s + (−0.5 + 0.866i)16-s + (0.0128 + 0.0221i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.991 + 0.132i)3-s + (−0.249 − 0.433i)4-s + (−0.448 − 0.258i)5-s + (−0.431 + 0.559i)6-s + (0.111 − 0.993i)7-s + 0.353·8-s + (0.964 + 0.263i)9-s + (0.316 − 0.182i)10-s + (−0.559 − 0.969i)11-s + (−0.190 − 0.462i)12-s + (0.100 − 0.994i)13-s + (0.568 + 0.419i)14-s + (−0.409 − 0.315i)15-s + (−0.125 + 0.216i)16-s + (0.00310 + 0.00537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.890 + 0.455i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42805 - 0.344038i\)
\(L(\frac12)\) \(\approx\) \(1.42805 - 0.344038i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.230i)T \)
7 \( 1 + (-0.296 + 2.62i)T \)
13 \( 1 + (-0.361 + 3.58i)T \)
good5 \( 1 + (1.00 + 0.578i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.85 + 3.21i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0128 - 0.0221i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.122 + 0.211i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 1.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.51iT - 29T^{2} \)
31 \( 1 + (-2.24 - 3.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.29 - 4.79i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.471iT - 41T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 + (0.203 + 0.117i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.44 - 3.72i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.80 - 1.61i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.2 - 5.91i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.24 - 3.60i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 + (-0.784 - 1.35i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.01 + 5.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.64iT - 83T^{2} \)
89 \( 1 + (8.11 + 4.68i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.44787604848212712838531586234, −9.822097344356546555689005834802, −8.664603777775978660784587933122, −8.018854425124146103021712888282, −7.54947968341762923976761393635, −6.35510057700756866951917549573, −5.03955654909343359827686614202, −4.01306579765940373694002892619, −2.92387174741680304687104322374, −0.909456548623422461847486382586, 1.85955046215319683431320676917, 2.71454484821187052712230096819, 3.89637063845067737285241889559, 4.95116795890337706862948733373, 6.61033208789187992255001622305, 7.58946975329063516329210572126, 8.276075922433014446957864341915, 9.294998313697630858439806942471, 9.657035441872187867586312963540, 10.90196114261762420901026698092

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