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Properties

Label 2-546-91.89-c1-0-16
Degree $2$
Conductor $546$
Sign $-0.738 + 0.674i$
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 + (−0.384 − 1.43i)5-s + (−0.258 − 0.965i)6-s + (−2.50 − 0.845i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.743 − 1.28i)10-s + (−0.000667 − 0.00248i)11-s + (0.500 − 0.866i)12-s + (−3.56 − 0.558i)13-s + (−1.17 − 2.37i)14-s + (−0.384 + 1.43i)15-s − 1.00·16-s − 6.55·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.499 − 0.288i)3-s + 0.500i·4-s + (−0.172 − 0.642i)5-s + (−0.105 − 0.394i)6-s + (−0.947 − 0.319i)7-s + (−0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.234 − 0.407i)10-s + (−0.000201 − 0.000750i)11-s + (0.144 − 0.250i)12-s + (−0.987 − 0.155i)13-s + (−0.314 − 0.633i)14-s + (−0.0993 + 0.370i)15-s − 0.250·16-s − 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.738 + 0.674i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.738 + 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130075 - 0.335296i\)
\(L(\frac12)\) \(\approx\) \(0.130075 - 0.335296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.845i)T \)
13 \( 1 + (3.56 + 0.558i)T \)
good5 \( 1 + (0.384 + 1.43i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.000667 + 0.00248i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + 6.55T + 17T^{2} \)
19 \( 1 + (1.62 + 0.435i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 0.506iT - 23T^{2} \)
29 \( 1 + (0.810 + 1.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (8.17 + 2.18i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.28 + 3.28i)T - 37iT^{2} \)
41 \( 1 + (-3.86 - 1.03i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.25 + 3.03i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.89 - 0.507i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.92 - 6.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.58 - 2.58i)T + 59iT^{2} \)
61 \( 1 + (-10.6 + 6.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.36 - 0.902i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (11.1 - 2.98i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.57 + 5.87i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.83 + 4.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.65 - 8.65i)T - 83iT^{2} \)
89 \( 1 + (-9.45 - 9.45i)T + 89iT^{2} \)
97 \( 1 + (4.02 + 15.0i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.57530647413863367785358953030, −9.459927757748309935572274417160, −8.653079211044494946129985449577, −7.45410729680034278856488579660, −6.79516886943068364505192872986, −5.88718532777025780359932113013, −4.83124362862594266056566346811, −4.00100239550124149347152661749, −2.43804315899719942659908620990, −0.17228088953095357988413024826, 2.28834417900387026798333861702, 3.36941421909846277295883204813, 4.45630532468042132405114950205, 5.49952037084526043742443695181, 6.57975189932012704420468054727, 7.11433065979307685639538401725, 8.800054891366939702736404782421, 9.616762515429742456875702812094, 10.42244298539905575319547820633, 11.17159445999130163520017811385

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