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Properties

Label 2-546-91.24-c1-0-11
Degree $2$
Conductor $546$
Sign $0.914 + 0.405i$
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 − i·3-s + (0.866 − 0.499i)4-s + (2.30 + 0.618i)5-s + (−0.258 − 0.965i)6-s + (1.78 + 1.94i)7-s + (0.707 − 0.707i)8-s − 9-s + 2.38·10-s + (−1.45 + 1.45i)11-s + (−0.499 − 0.866i)12-s + (3.60 + 0.0570i)13-s + (2.23 + 1.42i)14-s + (0.618 − 2.30i)15-s + (0.500 − 0.866i)16-s + (−1.37 − 2.38i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s − 0.577i·3-s + (0.433 − 0.249i)4-s + (1.03 + 0.276i)5-s + (−0.105 − 0.394i)6-s + (0.676 + 0.736i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s + 0.755·10-s + (−0.439 + 0.439i)11-s + (−0.144 − 0.249i)12-s + (0.999 + 0.0158i)13-s + (0.596 + 0.379i)14-s + (0.159 − 0.595i)15-s + (0.125 − 0.216i)16-s + (−0.334 − 0.579i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.914 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56430 - 0.542533i\)
\(L(\frac12)\) \(\approx\) \(2.56430 - 0.542533i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (-1.78 - 1.94i)T \)
13 \( 1 + (-3.60 - 0.0570i)T \)
good5 \( 1 + (-2.30 - 0.618i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.45 - 1.45i)T - 11iT^{2} \)
17 \( 1 + (1.37 + 2.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.53 - 4.53i)T - 19iT^{2} \)
23 \( 1 + (-2.12 - 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.69 + 2.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.60 + 5.97i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.80 + 10.4i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (9.91 + 2.65i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.84 - 3.95i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.0569 - 0.212i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.80 + 6.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.48 + 9.29i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 8.92iT - 61T^{2} \)
67 \( 1 + (-4.57 - 4.57i)T + 67iT^{2} \)
71 \( 1 + (5.69 - 1.52i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (11.7 - 3.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.79 - 6.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.18 - 9.18i)T - 83iT^{2} \)
89 \( 1 + (-0.410 + 0.110i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.25 - 4.66i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.94466157418907328093570271238, −10.01226659174498117076866720073, −8.967582896769682019924973891682, −8.014048803808460695220683079489, −6.90268113740731142291173562804, −5.89174257599817684215504099735, −5.45402855336360709800931336568, −4.03517458942810182860619530279, −2.45154075540142861583721976465, −1.80205889466226978028946076199, 1.65857508314816451596022188427, 3.17990408696155100362064651742, 4.38565906445386191868198956675, 5.15868787793410818441478841317, 6.09339800988768912135499961381, 7.00063673828063017819707568241, 8.408398551215214738000079303964, 8.906282933596179404156905471570, 10.37022493547126790683039100297, 10.69848559543727674849786277514

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