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Properties

Label 2-546-273.194-c1-0-28
Degree $2$
Conductor $546$
Sign $0.631 + 0.775i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46539 - 1.17160i\)
\(L(\frac12)\) \(\approx\) \(2.46539 - 1.17160i\)
\(L(\frac{3}{2})\) 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

−10.61369197401714548696126348551, −9.721550979189503323225893820924, −9.414438069382260767540390613272, −8.017086456281176114212306926381, −6.99143505148236732164403622350, −6.13574750733949932521206607143, −4.87211996601839146283838574872, −3.54770572011249087477888466151, −2.71426731280374709259835439109, −1.66129865430210128698516107530, 2.04745747004682002908511591565, 2.79384830364046137597557478771, 4.71231099104946135713992523186, 5.14452391528379098500256635371, 6.34067911193724131097043462129, 7.30092985964390548959718792442, 8.492989265470046485511156664385, 9.048164878686292933138373264409, 9.667922003287816911557706654221, 10.59538686372302778061284994424

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