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Properties

Label 2-546-91.51-c1-0-18
Degree $2$
Conductor $546$
Sign $-0.720 + 0.693i$
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  + (−0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (0.294 + 0.169i)5-s + 0.999i·6-s + (0.420 − 2.61i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.169 − 0.294i)10-s + (−0.571 + 0.330i)11-s + (0.499 − 0.866i)12-s + (0.660 − 3.54i)13-s + (−1.66 + 2.05i)14-s − 0.339i·15-s + (−0.5 + 0.866i)16-s + (3.27 + 5.66i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (0.131 + 0.0759i)5-s + 0.408i·6-s + (0.158 − 0.987i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.0537 − 0.0930i)10-s + (−0.172 + 0.0995i)11-s + (0.144 − 0.249i)12-s + (0.183 − 0.983i)13-s + (−0.446 + 0.548i)14-s − 0.0877i·15-s + (−0.125 + 0.216i)16-s + (0.793 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.720 + 0.693i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296885 - 0.735842i\)
\(L(\frac12)\) \(\approx\) \(0.296885 - 0.735842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.420 + 2.61i)T \)
13 \( 1 + (-0.660 + 3.54i)T \)
good5 \( 1 + (-0.294 - 0.169i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.571 - 0.330i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.27 - 5.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.27 + 3.04i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.71 + 6.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 + (-3.46 + 2i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.06 + 1.77i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.864iT - 41T^{2} \)
43 \( 1 + 5.08T + 43T^{2} \)
47 \( 1 + (8.18 + 4.72i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.38 - 1.95i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.932 - 1.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.45 + 3.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.884iT - 71T^{2} \)
73 \( 1 + (-8.72 + 5.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.22 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 + (-3.85 - 2.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−10.61585523491372950814731313085, −9.865811839898719707422329091994, −8.401041646166225740007442714833, −8.031334138575829540756194908187, −6.89446589311227127453879897089, −6.14284066759589232986473249716, −4.72489286943084607449331449332, −3.47744170240539154964307239828, −2.04246026006739435575504231147, −0.58191891545682600637432195264, 1.75241520128136213562531316942, 3.29795461025429757316418094837, 4.86918619435096282797047956465, 5.60510591746501061308238128253, 6.55814281653377241874415190888, 7.64417200730719357436910698735, 8.671401815333167922896752149928, 9.382848121577874140099155800034, 9.963582313934363726213939494841, 11.25612347021864638739768188436

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