Properties

Label 2-546-39.32-c1-0-17
Degree $2$
Conductor $546$
Sign $0.994 + 0.102i$
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 + (1.61 + 0.638i)3-s + (0.866 − 0.499i)4-s + (−0.489 − 0.489i)5-s + (1.72 + 0.199i)6-s + (0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (2.18 + 2.05i)9-s + (−0.599 − 0.345i)10-s + (0.247 + 0.923i)11-s + (1.71 − 0.252i)12-s + (2.67 + 2.41i)13-s i·14-s + (−0.475 − 1.09i)15-s + (0.500 − 0.866i)16-s + (−1.15 − 2.00i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.929 + 0.368i)3-s + (0.433 − 0.249i)4-s + (−0.218 − 0.218i)5-s + (0.702 + 0.0815i)6-s + (0.0978 − 0.365i)7-s + (0.249 − 0.249i)8-s + (0.728 + 0.685i)9-s + (−0.189 − 0.109i)10-s + (0.0745 + 0.278i)11-s + (0.494 − 0.0728i)12-s + (0.742 + 0.669i)13-s − 0.267i·14-s + (−0.122 − 0.283i)15-s + (0.125 − 0.216i)16-s + (−0.280 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.994 + 0.102i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.994 + 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.87404 - 0.147939i\)
\(L(\frac12)\) \(\approx\) \(2.87404 - 0.147939i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 0.638i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-2.67 - 2.41i)T \)
good5 \( 1 + (0.489 + 0.489i)T + 5iT^{2} \)
11 \( 1 + (-0.247 - 0.923i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.15 + 2.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.656 + 0.175i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.227 - 0.393i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.11 + 2.37i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.27 + 1.27i)T - 31iT^{2} \)
37 \( 1 + (-1.67 + 0.447i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (9.15 - 2.45i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.759 + 0.438i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.33 - 7.33i)T - 47iT^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + (11.3 + 3.04i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.88 - 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.53 + 9.44i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.951 + 3.55i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.44 + 4.44i)T + 73iT^{2} \)
79 \( 1 + 3.18T + 79T^{2} \)
83 \( 1 + (0.476 + 0.476i)T + 83iT^{2} \)
89 \( 1 + (0.780 + 2.91i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (11.6 + 3.12i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.83760549930628311966494166171, −9.905670714863189875434162219856, −9.075764192607969254278788579526, −8.128458510163010164263381341341, −7.22708199404724083753282696212, −6.17561170932159867855287949080, −4.70172459333197693142075911865, −4.15281630115718182372204747344, −3.04341649001600749061865186038, −1.73241137474838899826155329687, 1.76450025851933982231380194474, 3.13960513242232572719791431880, 3.81724602992770694311468098785, 5.22876360975350098659322530719, 6.32163505219106122805630400996, 7.16614354226553655174440708205, 8.192121846160436929406618595820, 8.713980253781043258401334647812, 9.916184631757900521126154632964, 10.97501998166623818692742411846

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