Properties

Label 2-546-91.59-c1-0-5
Degree $2$
Conductor $546$
Sign $0.358 - 0.933i$
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 + (−0.866 + 0.5i)3-s + 1.00i·4-s + (0.499 + 0.133i)5-s + (−0.965 − 0.258i)6-s + (0.430 − 2.61i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (0.258 + 0.447i)10-s + (4.55 + 1.21i)11-s + (−0.500 − 0.866i)12-s + (1.50 + 3.27i)13-s + (2.15 − 1.54i)14-s + (−0.499 + 0.133i)15-s − 1.00·16-s + 1.33·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.499 + 0.288i)3-s + 0.500i·4-s + (0.223 + 0.0598i)5-s + (−0.394 − 0.105i)6-s + (0.162 − 0.986i)7-s + (−0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (0.0817 + 0.141i)10-s + (1.37 + 0.367i)11-s + (−0.144 − 0.250i)12-s + (0.416 + 0.909i)13-s + (0.574 − 0.411i)14-s + (−0.128 + 0.0345i)15-s − 0.250·16-s + 0.324·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.358 - 0.933i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.358 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48337 + 1.01889i\)
\(L(\frac12)\) \(\approx\) \(1.48337 + 1.01889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.430 + 2.61i)T \)
13 \( 1 + (-1.50 - 3.27i)T \)
good5 \( 1 + (-0.499 - 0.133i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-4.55 - 1.21i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 + (-2.16 - 8.08i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.09iT - 23T^{2} \)
29 \( 1 + (1.91 - 3.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.785 - 2.93i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.49 - 1.49i)T - 37iT^{2} \)
41 \( 1 + (1.92 + 7.18i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.73 + 5.04i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.67 - 6.25i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.49 + 4.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.738 + 0.738i)T + 59iT^{2} \)
61 \( 1 + (-9.64 - 5.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.31 + 4.89i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.85 + 6.93i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.00 + 0.804i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.47 + 3.47i)T - 83iT^{2} \)
89 \( 1 + (4.91 + 4.91i)T + 89iT^{2} \)
97 \( 1 + (14.7 + 3.94i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.00270849599641062548564699813, −10.12669908886564962145561683726, −9.286322693747030045836793908734, −8.160868642403069702793181944341, −7.05715854335880701180507304227, −6.44029478680173748064598817265, −5.46980861422891029649526563312, −4.16248886628858094131891275434, −3.78100701354446467461148946517, −1.54348984262542430608517120585, 1.15089126008992735807812363790, 2.58729712105676007054118760112, 3.82938809379102640201886792907, 5.19371461236152954413995712667, 5.81484519874154109019254194141, 6.70349826769788303096444104020, 7.964769066629710694293453331912, 9.155536404900292400819324536096, 9.676428427442807018781602662714, 11.09680749822164569999243000538

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