Properties

Label 2-546-91.59-c1-0-10
Degree $2$
Conductor $546$
Sign $-0.215 + 0.976i$
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 + (−1.30 − 0.349i)5-s + (0.965 + 0.258i)6-s + (−1.83 + 1.90i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.675 + 1.16i)10-s + (2.57 + 0.689i)11-s + (−0.500 − 0.866i)12-s + (−3.33 − 1.36i)13-s + (2.64 − 0.0509i)14-s + (1.30 − 0.349i)15-s − 1.00·16-s + 5.24·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.499 + 0.288i)3-s + 0.500i·4-s + (−0.583 − 0.156i)5-s + (0.394 + 0.105i)6-s + (−0.693 + 0.720i)7-s + (0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (0.213 + 0.369i)10-s + (0.775 + 0.207i)11-s + (−0.144 − 0.250i)12-s + (−0.925 − 0.379i)13-s + (0.706 − 0.0136i)14-s + (0.336 − 0.0902i)15-s − 0.250·16-s + 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.215 + 0.976i$
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: \((2,\ 546,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331544 - 0.412863i\)
\(L(\frac12)\) \(\approx\) \(0.331544 - 0.412863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 - 1.90i)T \)
13 \( 1 + (3.33 + 1.36i)T \)
good5 \( 1 + (1.30 + 0.349i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.57 - 0.689i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 + (1.78 + 6.66i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.262iT - 23T^{2} \)
29 \( 1 + (-3.07 + 5.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.112 + 0.421i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.94 + 1.94i)T - 37iT^{2} \)
41 \( 1 + (2.81 + 10.5i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.74 + 3.89i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.48 - 5.55i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.878 + 1.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.09 + 6.09i)T + 59iT^{2} \)
61 \( 1 + (7.62 + 4.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.68 - 10.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.08 + 4.04i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.86 + 1.30i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.12 - 5.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.32 + 9.32i)T - 83iT^{2} \)
89 \( 1 + (-4.38 - 4.38i)T + 89iT^{2} \)
97 \( 1 + (18.5 + 4.97i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.50235436733119770389747128444, −9.627885165980127046223984791630, −9.102878816642725670181063734636, −7.962696207120044991399038177327, −7.02334756305371005631016436514, −5.94710121169077358525745778187, −4.77288991421136306073950701784, −3.67452962928099683564883713235, −2.47143656568227808358829674658, −0.42770223745161770309365823612, 1.29269480041761877403102107237, 3.36762140655425861335971770307, 4.49161297564806428567776670250, 5.83753320785919765116105457941, 6.62223681735868478759953898080, 7.46503225545660306903153017950, 8.091753665127692874349788908713, 9.440159910078327785461843214306, 10.07626669481246292075531885443, 10.90761186268877778102358709326

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