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Properties

Label 2-546-273.38-c1-0-20
Degree $2$
Conductor $546$
Sign $0.553 + 0.832i$
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 + (1.73 + 0.0505i)3-s + (−0.499 + 0.866i)4-s + (−3.26 + 1.88i)5-s + (−0.821 − 1.52i)6-s + (0.385 − 2.61i)7-s + 0.999·8-s + (2.99 + 0.175i)9-s + (3.26 + 1.88i)10-s + (1.89 − 3.27i)11-s + (−0.909 + 1.47i)12-s + (2.90 − 2.13i)13-s + (−2.45 + 0.975i)14-s + (−5.74 + 3.09i)15-s + (−0.5 − 0.866i)16-s + (−1.16 + 2.00i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.999 + 0.0292i)3-s + (−0.249 + 0.433i)4-s + (−1.45 + 0.842i)5-s + (−0.335 − 0.622i)6-s + (0.145 − 0.989i)7-s + 0.353·8-s + (0.998 + 0.0584i)9-s + (1.03 + 0.595i)10-s + (0.570 − 0.988i)11-s + (−0.262 + 0.425i)12-s + (0.805 − 0.593i)13-s + (−0.657 + 0.260i)14-s + (−1.48 + 0.799i)15-s + (−0.125 − 0.216i)16-s + (−0.281 + 0.487i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25452 - 0.672655i\)
\(L(\frac12)\) \(\approx\) \(1.25452 - 0.672655i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 0.0505i)T \)
7 \( 1 + (-0.385 + 2.61i)T \)
13 \( 1 + (-2.90 + 2.13i)T \)
good5 \( 1 + (3.26 - 1.88i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.89 + 3.27i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.16 - 2.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.00 - 5.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.79 + 3.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.07iT - 29T^{2} \)
31 \( 1 + (-4.94 + 8.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.92 - 3.42i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.69iT - 41T^{2} \)
43 \( 1 + 3.47T + 43T^{2} \)
47 \( 1 + (-6.71 + 3.87i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.653 + 0.377i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.68 - 2.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.37 - 4.83i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.51 + 2.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.23T + 71T^{2} \)
73 \( 1 + (-2.53 + 4.38i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.402 + 0.696i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.88iT - 83T^{2} \)
89 \( 1 + (13.5 - 7.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.04T + 97T^{2} \)
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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

−10.68927550484193913351115317910, −9.988676970422919900323841353296, −8.652298222088532262711285399551, −8.098930715917882823452905198715, −7.46652982496831987217675028057, −6.43974042962149573356259092272, −4.28899286260260724880828460193, −3.64397782818361594220311613754, −3.01189507345573744503395065488, −1.03089238225900138594532902414, 1.41361360765885641204413188821, 3.19301402839231419326829742880, 4.41996316752550555310744628317, 5.09715610758260829046777039566, 6.94643698253212635478866861011, 7.33043921416739111584874726059, 8.589625326876407823889542099121, 8.844025019280194624358033323632, 9.459225728080955066208656780932, 10.99582657115824163982765975696

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