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Properties

Label 2-546-91.5-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.603 + 0.797i$
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 + (0.866 − 0.5i)3-s + (−0.866 + 0.499i)4-s + (−4.17 + 1.11i)5-s + (0.707 + 0.707i)6-s + (−0.344 + 2.62i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−2.16 − 3.74i)10-s + (1.30 − 4.88i)11-s + (−0.5 + 0.866i)12-s + (−3.28 − 1.48i)13-s + (−2.62 + 0.346i)14-s + (−3.05 + 3.05i)15-s + (0.500 − 0.866i)16-s + (−1.53 − 2.65i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.499 − 0.288i)3-s + (−0.433 + 0.249i)4-s + (−1.86 + 0.500i)5-s + (0.288 + 0.288i)6-s + (−0.130 + 0.991i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.683 − 1.18i)10-s + (0.394 − 1.47i)11-s + (−0.144 + 0.249i)12-s + (−0.911 − 0.410i)13-s + (−0.701 + 0.0925i)14-s + (−0.789 + 0.789i)15-s + (0.125 − 0.216i)16-s + (−0.372 − 0.644i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.603 + 0.797i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00525304 - 0.0105716i\)
\(L(\frac12)\) \(\approx\) \(0.00525304 - 0.0105716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.344 - 2.62i)T \)
13 \( 1 + (3.28 + 1.48i)T \)
good5 \( 1 + (4.17 - 1.11i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.30 + 4.88i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.53 + 2.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.43 - 0.919i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.27 + 1.89i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.32T + 29T^{2} \)
31 \( 1 + (2.08 - 7.76i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.74 - 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.16 + 2.16i)T + 41iT^{2} \)
43 \( 1 - 8.74iT - 43T^{2} \)
47 \( 1 + (-2.31 - 8.63i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.196 + 0.339i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.810 - 0.217i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.31 - 1.91i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.85 + 1.57i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.51 + 1.51i)T - 71iT^{2} \)
73 \( 1 + (2.38 + 0.638i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.16 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.36 + 6.36i)T + 83iT^{2} \)
89 \( 1 + (-1.74 - 6.53i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.94 - 9.94i)T + 97iT^{2} \)
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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

−10.65163395948772011402840359047, −9.130311985794085947326295346904, −8.474582388117993893254377459387, −7.85318362654856461136545669106, −6.99545090411833861321629077577, −6.08671293949607239282189123400, −4.76242769839837520676820395895, −3.60133259370392802769915312930, −2.85149575108296106764828372426, −0.00584847414334260869560658981, 2.00714469619496441689938862450, 3.89987000441426299892647668496, 3.99515030473306046867113657370, 4.91117063048706447119025629829, 7.01414705278788696399606728720, 7.54847855808264975788173850950, 8.507374658266784328244752333506, 9.454935718451422135179355277446, 10.28130806298257191892395690870, 11.19872294393221788732161283114

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