Properties

Label 2-546-273.200-c1-0-18
Degree $2$
Conductor $546$
Sign $0.821 + 0.570i$
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  + (−0.258 + 0.965i)2-s + (−1.11 − 1.32i)3-s + (−0.866 − 0.499i)4-s + (0.488 − 1.82i)5-s + (1.56 − 0.737i)6-s + (0.619 + 2.57i)7-s + (0.707 − 0.707i)8-s + (−0.500 + 2.95i)9-s + (1.63 + 0.944i)10-s + (0.458 + 1.71i)11-s + (0.306 + 1.70i)12-s + (1.62 − 3.22i)13-s + (−2.64 − 0.0671i)14-s + (−2.96 + 1.39i)15-s + (0.500 + 0.866i)16-s + (2.01 − 3.48i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.645 − 0.763i)3-s + (−0.433 − 0.249i)4-s + (0.218 − 0.816i)5-s + (0.639 − 0.301i)6-s + (0.234 + 0.972i)7-s + (0.249 − 0.249i)8-s + (−0.166 + 0.985i)9-s + (0.517 + 0.298i)10-s + (0.138 + 0.516i)11-s + (0.0885 + 0.492i)12-s + (0.449 − 0.893i)13-s + (−0.706 − 0.0179i)14-s + (−0.764 + 0.359i)15-s + (0.125 + 0.216i)16-s + (0.488 − 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.821 + 0.570i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.821 + 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00784 - 0.315606i\)
\(L(\frac12)\) \(\approx\) \(1.00784 - 0.315606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (1.11 + 1.32i)T \)
7 \( 1 + (-0.619 - 2.57i)T \)
13 \( 1 + (-1.62 + 3.22i)T \)
good5 \( 1 + (-0.488 + 1.82i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.458 - 1.71i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.01 + 3.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.18 - 0.854i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.75 + 6.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.36iT - 29T^{2} \)
31 \( 1 + (2.00 + 7.49i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.92 + 7.18i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.16 + 1.16i)T + 41iT^{2} \)
43 \( 1 + 1.15iT - 43T^{2} \)
47 \( 1 + (-9.54 - 2.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.17 - 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.26 - 12.1i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.96 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.84 - 10.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.54 - 1.54i)T + 71iT^{2} \)
73 \( 1 + (1.64 - 0.439i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.34 + 5.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.17 - 4.17i)T + 83iT^{2} \)
89 \( 1 + (3.18 + 0.853i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.14 + 5.14i)T - 97iT^{2} \)
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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

−10.71057873675588617243552113004, −9.641972343743745123321989802425, −8.764180060574659605647770118673, −7.954507874463296562541326736177, −7.18305300963567377600652442369, −5.79968323217994955171234890658, −5.61121460519222845924872183133, −4.48079845573827567029029713357, −2.37607545622706325906310755559, −0.833719840949430176651221624348, 1.34106454451574783755860183016, 3.31216994297274645372455270615, 3.92246257003636965655712673483, 5.14708717430225848387785740328, 6.28625595819105376131998846288, 7.18847604904663438333611780500, 8.449707955826844266705639004578, 9.493068321793268349982434918991, 10.25330755851842758861089955715, 10.80600261163434522901771139061

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