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Properties

Label 2-546-273.38-c1-0-25
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: $\chi_{546} (311, \cdot )$
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

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

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