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Properties

Label 2-546-91.20-c1-0-14
Degree $2$
Conductor $546$
Sign $0.966 + 0.255i$
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 + (1.14 − 1.14i)5-s + (0.965 + 0.258i)6-s + (2.56 + 0.651i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.806 − 1.39i)10-s + (−0.325 − 1.21i)11-s + 12-s + (−3.51 + 0.808i)13-s + (2.64 − 0.0342i)14-s + (1.55 − 0.417i)15-s + (0.500 − 0.866i)16-s + (0.338 + 0.586i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (0.510 − 0.510i)5-s + (0.394 + 0.105i)6-s + (0.969 + 0.246i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (0.255 − 0.441i)10-s + (−0.0982 − 0.366i)11-s + 0.288·12-s + (−0.974 + 0.224i)13-s + (0.707 − 0.00916i)14-s + (0.402 − 0.107i)15-s + (0.125 − 0.216i)16-s + (0.0820 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.966 + 0.255i$
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: \((2,\ 546,\ (\ :1/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.81074 - 0.365622i\)
\(L(\frac12)\) \(\approx\) \(2.81074 - 0.365622i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 0.651i)T \)
13 \( 1 + (3.51 - 0.808i)T \)
good5 \( 1 + (-1.14 + 1.14i)T - 5iT^{2} \)
11 \( 1 + (0.325 + 1.21i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.338 - 0.586i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.48 + 1.46i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.30 - 1.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.92 + 3.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.505 - 0.505i)T - 31iT^{2} \)
37 \( 1 + (1.16 + 4.34i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.69 - 6.34i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.12 - 0.652i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.433 + 0.433i)T + 47iT^{2} \)
53 \( 1 + 4.52T + 53T^{2} \)
59 \( 1 + (-1.06 + 3.97i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.66 + 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.6 - 3.65i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.97 - 11.1i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.598 - 0.598i)T + 73iT^{2} \)
79 \( 1 + 9.59T + 79T^{2} \)
83 \( 1 + (-0.963 + 0.963i)T - 83iT^{2} \)
89 \( 1 + (-8.80 + 2.35i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (13.9 + 3.73i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.88174730729253599725074190058, −9.903783692084287041428047975825, −9.013887248294756177633093012651, −8.199959350531137096702634839987, −7.15712463161643489311979877639, −5.86705329472800399637610009426, −4.96898545272285139796264487734, −4.26314067595900115513736947194, −2.76626962356476814701385808601, −1.71958078128018882174401657686, 1.89070910710784085733486444225, 2.82395075009576667327716123579, 4.26654531443455118435221622471, 5.12778953076782759413434906409, 6.34451709473234144309513829239, 7.18656946283241039372930238749, 7.963213918298421279081302543006, 8.921243542923880550374913831890, 10.21221816598233685736103873046, 10.73095668121196712260397440132

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