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Properties

Label 2-546-91.80-c1-0-11
Degree $2$
Conductor $546$
Sign $0.734 + 0.678i$
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 − i·3-s + (−0.866 + 0.499i)4-s + (0.711 − 2.65i)5-s + (0.965 − 0.258i)6-s + (2.33 + 1.23i)7-s + (−0.707 − 0.707i)8-s − 9-s + 2.74·10-s + (−1.56 − 1.56i)11-s + (0.499 + 0.866i)12-s + (0.354 − 3.58i)13-s + (−0.589 + 2.57i)14-s + (−2.65 − 0.711i)15-s + (0.500 − 0.866i)16-s + (−0.838 − 1.45i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s − 0.577i·3-s + (−0.433 + 0.249i)4-s + (0.318 − 1.18i)5-s + (0.394 − 0.105i)6-s + (0.883 + 0.467i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s + 0.868·10-s + (−0.472 − 0.472i)11-s + (0.144 + 0.249i)12-s + (0.0983 − 0.995i)13-s + (−0.157 + 0.689i)14-s + (−0.685 − 0.183i)15-s + (0.125 − 0.216i)16-s + (−0.203 − 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49589 - 0.585368i\)
\(L(\frac12)\) \(\approx\) \(1.49589 - 0.585368i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (-2.33 - 1.23i)T \)
13 \( 1 + (-0.354 + 3.58i)T \)
good5 \( 1 + (-0.711 + 2.65i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.56 + 1.56i)T + 11iT^{2} \)
17 \( 1 + (0.838 + 1.45i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.129 - 0.129i)T + 19iT^{2} \)
23 \( 1 + (-0.472 - 0.273i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.18 + 3.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-7.55 + 2.02i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.24 - 0.333i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.32 + 4.95i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.86 + 1.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.74 - 2.61i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.78 + 3.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.43 + 0.384i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 13.6iT - 61T^{2} \)
67 \( 1 + (4.98 - 4.98i)T - 67iT^{2} \)
71 \( 1 + (-1.04 - 3.91i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.29 - 8.57i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.42 - 7.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.3 + 11.3i)T + 83iT^{2} \)
89 \( 1 + (-4.16 - 15.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.6 - 4.18i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.75563767664644933127112783836, −9.519558744649577395587296683530, −8.499901118419664670399946543040, −8.213497689173619599652696603878, −7.19685705159572619943947639183, −5.75261660196860482030131878138, −5.42659541333913957016003629505, −4.34938672922714883167505624335, −2.58740766209695261855830620922, −0.945757469205167160653286549820, 1.85507985139712734344869058567, 3.00201902363909505766169648317, 4.18919360005673831206261069581, 4.99287510912940649702875432796, 6.30125939690204729614623896072, 7.26026856599896992724696886018, 8.391299593154434532796517042754, 9.449011320075148860013602091534, 10.32787555775686925555472060774, 10.81214968020038840549791216905

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