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Properties

Label 2-546-91.20-c1-0-19
Degree $2$
Conductor $546$
Sign $-0.0472 + 0.998i$
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 + (1.73 − 1.73i)5-s + (−0.965 − 0.258i)6-s + (0.130 − 2.64i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (1.22 − 2.12i)10-s + (0.282 + 1.05i)11-s − 12-s + (−3.50 + 0.850i)13-s + (−0.557 − 2.58i)14-s + (−2.36 + 0.633i)15-s + (0.500 − 0.866i)16-s + (−2.90 − 5.03i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (0.774 − 0.774i)5-s + (−0.394 − 0.105i)6-s + (0.0493 − 0.998i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.387 − 0.670i)10-s + (0.0851 + 0.317i)11-s − 0.288·12-s + (−0.971 + 0.235i)13-s + (−0.149 − 0.691i)14-s + (−0.610 + 0.163i)15-s + (0.125 − 0.216i)16-s + (−0.704 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0472 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0472 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0472 + 0.998i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0472 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37632 - 1.44298i\)
\(L(\frac12)\) \(\approx\) \(1.37632 - 1.44298i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.130 + 2.64i)T \)
13 \( 1 + (3.50 - 0.850i)T \)
good5 \( 1 + (-1.73 + 1.73i)T - 5iT^{2} \)
11 \( 1 + (-0.282 - 1.05i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.90 + 5.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.84 - 1.02i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.76 - 1.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.20 - 2.08i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.473 - 0.473i)T - 31iT^{2} \)
37 \( 1 + (2.75 + 10.2i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.179 + 0.671i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.63 + 4.40i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.26 - 8.26i)T + 47iT^{2} \)
53 \( 1 - 6.15T + 53T^{2} \)
59 \( 1 + (1.41 - 5.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.34 - 1.35i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.50 + 2.27i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.02 - 11.2i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.95 - 7.95i)T + 73iT^{2} \)
79 \( 1 - 0.579T + 79T^{2} \)
83 \( 1 + (7.84 - 7.84i)T - 83iT^{2} \)
89 \( 1 + (10.5 - 2.82i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.85 + 1.83i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.74156805882925783707783807601, −9.738431439852513860960348171804, −9.112049473414858972725305152521, −7.33323149999919470498272664933, −7.10201402607052376420464962633, −5.63341059547193488238352164685, −5.03582272051811832561719310199, −4.08093308096493311706290408083, −2.40635945943453012977757628066, −1.03951674850640659942976546385, 2.17334430807319420414925112320, 3.16419312900648164305041086755, 4.63254712986426616914043235030, 5.58174216130559090554238353102, 6.22402558302120503930153209979, 7.06328621243434521946722178053, 8.351497974633895498122729464248, 9.431282926775066663366971779900, 10.29096435153413568296442228409, 11.07878315034822802154650816680

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