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Properties

Label 2-546-91.73-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.316 - 0.948i$
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  + (−0.258 + 0.965i)2-s + (0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (3.05 + 0.819i)5-s + (−0.707 + 0.707i)6-s + (−2.47 + 0.925i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.58 + 2.74i)10-s + (−0.0343 − 0.128i)11-s + (−0.5 − 0.866i)12-s + (−0.467 + 3.57i)13-s + (−0.252 − 2.63i)14-s + (2.23 + 2.23i)15-s + (0.500 + 0.866i)16-s + (−1.61 + 2.80i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (1.36 + 0.366i)5-s + (−0.288 + 0.288i)6-s + (−0.936 + 0.349i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.500 + 0.867i)10-s + (−0.0103 − 0.0386i)11-s + (−0.144 − 0.249i)12-s + (−0.129 + 0.991i)13-s + (−0.0676 − 0.703i)14-s + (0.578 + 0.578i)15-s + (0.125 + 0.216i)16-s + (−0.392 + 0.679i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967418 + 1.34256i\)
\(L(\frac12)\) \(\approx\) \(0.967418 + 1.34256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (2.47 - 0.925i)T \)
13 \( 1 + (0.467 - 3.57i)T \)
good5 \( 1 + (-3.05 - 0.819i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0343 + 0.128i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.61 - 2.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.22 - 0.864i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.82 + 1.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.41T + 29T^{2} \)
31 \( 1 + (0.494 + 1.84i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.30 - 1.69i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.28 - 4.28i)T - 41iT^{2} \)
43 \( 1 + 6.94iT - 43T^{2} \)
47 \( 1 + (-2.06 + 7.68i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.30 - 7.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.58 - 1.76i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.51 - 5.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.2 + 3.29i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.99 + 7.99i)T + 71iT^{2} \)
73 \( 1 + (-6.08 + 1.62i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.51 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.334 - 0.334i)T - 83iT^{2} \)
89 \( 1 + (-4.46 + 16.6i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.12 + 4.12i)T - 97iT^{2} \)
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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

−10.62877615081619859428194777148, −9.900147643079395162788787185774, −9.308841578970573460276680651315, −8.670864194883516116505376249033, −7.33428736534385858236627287817, −6.40679204436254043520779312569, −5.85157366990998257003314459563, −4.59559352578286910342530504170, −3.18167744527234806688257141921, −1.95466402580129332227068893923, 1.02432702352147518271369741407, 2.47819422765900936151882993614, 3.29595289331190621686188965688, 4.85022885428785778262934202428, 5.89639424778610277877285001887, 6.94391786893138771332130636641, 8.015056628330004924840479927880, 9.201537182773080538121393027127, 9.587457926448186214200377881258, 10.27866425236651217106952364661

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