Properties

Label 2-546-273.32-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.814 - 0.579i$
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  + (−0.707 − 0.707i)2-s + (−0.222 + 1.71i)3-s + 1.00i·4-s + (−0.998 + 3.72i)5-s + (1.37 − 1.05i)6-s + (0.528 + 2.59i)7-s + (0.707 − 0.707i)8-s + (−2.90 − 0.762i)9-s + (3.34 − 1.92i)10-s + (3.62 + 0.970i)11-s + (−1.71 − 0.222i)12-s + (3.53 − 0.704i)13-s + (1.45 − 2.20i)14-s + (−6.17 − 2.54i)15-s − 1.00·16-s − 0.858·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.128 + 0.991i)3-s + 0.500i·4-s + (−0.446 + 1.66i)5-s + (0.559 − 0.431i)6-s + (0.199 + 0.979i)7-s + (0.250 − 0.250i)8-s + (−0.967 − 0.254i)9-s + (1.05 − 0.609i)10-s + (1.09 + 0.292i)11-s + (−0.495 − 0.0641i)12-s + (0.980 − 0.195i)13-s + (0.390 − 0.589i)14-s + (−1.59 − 0.656i)15-s − 0.250·16-s − 0.208·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.814 - 0.579i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.814 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284147 + 0.889964i\)
\(L(\frac12)\) \(\approx\) \(0.284147 + 0.889964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.222 - 1.71i)T \)
7 \( 1 + (-0.528 - 2.59i)T \)
13 \( 1 + (-3.53 + 0.704i)T \)
good5 \( 1 + (0.998 - 3.72i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.62 - 0.970i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 0.858T + 17T^{2} \)
19 \( 1 + (-1.11 - 4.15i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.07T + 23T^{2} \)
29 \( 1 + (-2.97 - 1.71i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.42 + 5.31i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.06 + 4.06i)T + 37iT^{2} \)
41 \( 1 + (-0.774 + 0.207i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.15 + 0.664i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.65 - 1.51i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.07 - 2.35i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.2 + 10.2i)T - 59iT^{2} \)
61 \( 1 + (1.55 - 2.68i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.5 + 3.62i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.326 + 1.21i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.80 + 1.28i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.06 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.70 - 4.70i)T + 83iT^{2} \)
89 \( 1 + (0.827 - 0.827i)T - 89iT^{2} \)
97 \( 1 + (3.55 + 0.953i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.04607980875874096077343719599, −10.37786151158370805531743709044, −9.570228235986702996849086272264, −8.703924716811269542964597327919, −7.80023512909478181770211563765, −6.52564153801347648679950720030, −5.78055197303797925882886688865, −4.03638932573418887785729924963, −3.44576179165604127395160944981, −2.24900397852564139863555950860, 0.71295648623898215487321079981, 1.49206913896980466603259681278, 3.88389901196089377366299415939, 4.90318701451331659660818581124, 6.05237210998630545489282318437, 6.94183777218946299307684551791, 7.83672286564554860707003504427, 8.679973392655368617607620132181, 9.014809906333992723525583551789, 10.44382059269269503797572301605

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