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Properties

Label 2-546-91.47-c1-0-8
Degree $2$
Conductor $546$
Sign $0.920 - 0.389i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 − 0.499i)4-s + (0.819 + 3.05i)5-s + (0.707 − 0.707i)6-s + (−0.925 + 2.47i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (1.58 + 2.74i)10-s + (0.128 + 0.0343i)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  + (0.683 − 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (0.366 + 1.36i)5-s + (0.288 − 0.288i)6-s + (−0.349 + 0.936i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.500 + 0.867i)10-s + (0.0386 + 0.0103i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.920 - 0.389i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.920 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55047 + 0.517767i\)
\(L(\frac12)\) \(\approx\) \(2.55047 + 0.517767i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.67392046497852236994169325279, −10.29126844067874119717892941779, −9.180270273177731902363911395734, −8.040432627129360754859506766725, −7.11763758444909537177363189405, −6.10464795378027217619690687331, −5.59255128394208012250402370871, −3.76151034305006037330047311130, −2.95766587560894303572528120099, −2.06774886680738780857516397196, 1.36185802257820311834404023153, 3.03199284005900412145873001388, 4.34119897172293017521385008479, 4.77587439635630370027055981304, 6.03452214135836053589251753722, 7.12284102281721907682860015909, 8.034981589162658405338055056403, 9.141165046061620422424128725407, 9.608164755317520870920976194908, 10.76827715592900784190702173667

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