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Properties

Label 2-546-7.4-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.266 - 0.963i$
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s + 0.999·6-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.999 + 1.73i)10-s + (−1.5 + 2.59i)11-s + (0.499 + 0.866i)12-s − 13-s + (−2.5 + 0.866i)14-s + 1.99·15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.447 + 0.774i)5-s + 0.408·6-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (−0.452 + 0.783i)11-s + (0.144 + 0.249i)12-s − 0.277·13-s + (−0.668 + 0.231i)14-s + 0.516·15-s + (−0.125 − 0.216i)16-s + (−0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07031 + 1.40690i\)
\(L(\frac12)\) \(\approx\) \(1.07031 + 1.40690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
13 \( 1 + T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + (-6 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−11.11627975423796464173872017574, −9.992170014811414547373982348184, −9.253347824562948930149468558988, −8.160893328886625656840873248713, −7.41041319569221088915125316641, −6.41941987325533303280365592014, −5.81242054577302468369073997872, −4.60167290071852992059786229672, −3.05306367969322563837806220540, −2.20043645908680964663657032609, 0.911323824910058518842134785462, 2.63931602917772875355829287061, 3.75994407612231324242106755751, 4.78112701832998446623982469398, 5.52874986029181995125765103422, 6.85657401507790752482144449679, 8.084624562723368102394740189153, 9.039277001127544061606992002705, 9.731708604252730298311664446684, 10.58948785133078455353567380320

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