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Properties

Label 2-546-7.4-c1-0-10
Degree $2$
Conductor $546$
Sign $0.931 + 0.364i$
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 + (−1.30 − 2.26i)5-s + 0.999·6-s + (1.77 + 1.95i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.30 − 2.26i)10-s + (1.77 − 3.08i)11-s + (0.499 + 0.866i)12-s + 13-s + (−0.806 + 2.51i)14-s − 2.61·15-s + (−0.5 − 0.866i)16-s + (3.97 − 6.88i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.584 − 1.01i)5-s + 0.408·6-s + (0.672 + 0.740i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.413 − 0.715i)10-s + (0.536 − 0.929i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (−0.215 + 0.673i)14-s − 0.674·15-s + (−0.125 − 0.216i)16-s + (0.964 − 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.931 + 0.364i$
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: \((2,\ 546,\ (\ :1/2),\ 0.931 + 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79746 - 0.339334i\)
\(L(\frac12)\) \(\approx\) \(1.79746 - 0.339334i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.77 - 1.95i)T \)
13 \( 1 - T \)
good5 \( 1 + (1.30 + 2.26i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.77 + 3.08i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.97 + 6.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.08 - 1.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.27 - 2.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-0.414 + 0.717i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.89 + 6.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 0.773T + 43T^{2} \)
47 \( 1 + (-1.47 - 2.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.28 - 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.39 - 4.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.36 + 5.82i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.25 - 12.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.82T + 71T^{2} \)
73 \( 1 + (5.27 - 9.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.19 - 2.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + (2.10 + 3.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−11.11528265998350990427414807716, −9.249552254833200396446797703181, −8.935138428452857564993020307798, −7.912151073363825105650732267670, −7.43275636453549136814637379778, −5.95009963967469842967297547834, −5.31199508667136995157043206532, −4.19075974935540139489128494179, −2.95545909669024976087398218715, −1.07250534433156500902254939539, 1.68133929066394928891195262419, 3.23968560466899010679009961817, 3.96285091442084637246653561786, 4.84328553146004376769497771705, 6.27659396011283165035589279879, 7.33934610710358968093364410885, 8.137871819110295681236988840341, 9.324480482809467995739554270722, 10.37084494221813223005474303480, 10.71429082063883570979683203792

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