Properties

Label 2-546-91.33-c1-0-12
Degree $2$
Conductor $546$
Sign $0.964 + 0.264i$
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 + (0.379 + 1.41i)5-s + (0.965 + 0.258i)6-s + (1.63 − 2.07i)7-s + (0.707 − 0.707i)8-s − 9-s − 1.46·10-s + (2.67 − 2.67i)11-s + (−0.499 + 0.866i)12-s + (−3.50 + 0.851i)13-s + (1.58 + 2.11i)14-s + (1.41 − 0.379i)15-s + (0.500 + 0.866i)16-s + (3.53 − 6.12i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s − 0.577i·3-s + (−0.433 − 0.249i)4-s + (0.169 + 0.633i)5-s + (0.394 + 0.105i)6-s + (0.618 − 0.785i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s − 0.463·10-s + (0.805 − 0.805i)11-s + (−0.144 + 0.249i)12-s + (−0.971 + 0.236i)13-s + (0.423 + 0.566i)14-s + (0.365 − 0.0980i)15-s + (0.125 + 0.216i)16-s + (0.857 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.964 + 0.264i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.964 + 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34907 - 0.181838i\)
\(L(\frac12)\) \(\approx\) \(1.34907 - 0.181838i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 2.07i)T \)
13 \( 1 + (3.50 - 0.851i)T \)
good5 \( 1 + (-0.379 - 1.41i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.67 + 2.67i)T - 11iT^{2} \)
17 \( 1 + (-3.53 + 6.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.44 - 3.44i)T - 19iT^{2} \)
23 \( 1 + (-1.90 + 1.09i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.51 + 2.63i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.79 - 2.35i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.18 + 1.12i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.520 - 1.94i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.52 + 2.03i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.6 + 2.85i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.155 + 0.268i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.55 - 0.683i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 2.01iT - 61T^{2} \)
67 \( 1 + (7.39 + 7.39i)T + 67iT^{2} \)
71 \( 1 + (0.304 - 1.13i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.17 - 11.8i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.15 - 14.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + (0.587 - 2.19i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.84 - 1.03i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.64725423995477482459992617234, −9.894325436689599366960984847665, −8.785139486870695016858438747924, −7.903339229975233380976340162126, −7.08388175978650674235491860093, −6.51155613680067196224448479245, −5.35061725586119680082201941215, −4.21192438911639057723791465063, −2.73651774014498251590629701752, −0.969152623184395793613803809646, 1.51799717982307527564874727593, 2.78986004377686387329115723110, 4.29960278519754200168290869905, 4.90686586883680068113238005263, 6.00117598371332728865609887895, 7.48501514023508200037224746471, 8.633191148409600824780088532017, 9.046836055782810163259984285909, 10.01976331215566421890673478562, 10.69863105913270271259098346528

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