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Properties

Label 2-546-13.12-c1-0-2
Degree $2$
Conductor $546$
Sign $0.832 - 0.554i$
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  − i·2-s + 3-s − 4-s + 3i·5-s − i·6-s − i·7-s + i·8-s + 9-s + 3·10-s + 5i·11-s − 12-s + (−3 + 2i)13-s − 14-s + 3i·15-s + 16-s + 3·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.34i·5-s − 0.408i·6-s − 0.377i·7-s + 0.353i·8-s + 0.333·9-s + 0.948·10-s + 1.50i·11-s − 0.288·12-s + (−0.832 + 0.554i)13-s − 0.267·14-s + 0.774i·15-s + 0.250·16-s + 0.727·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47541 + 0.446720i\)
\(L(\frac12)\) \(\approx\) \(1.47541 + 0.446720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - T \)
7 \( 1 + iT \)
13 \( 1 + (3 - 2i)T \)
good5 \( 1 - 3iT - 5T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−10.64424565668071911071373429551, −10.03045588248703020413097068827, −9.566490198455084588934479518599, −8.179094225529055222791246943536, −7.26015157601889847259167581047, −6.65867049724933434258730447769, −4.96054312587536855532952426758, −3.92534040678059442665584062182, −2.86521533116791941437839235681, −1.92143899985575659444802638883, 0.881998858320457016092041279158, 2.86349550330828889337040673671, 4.18675780044876769409520304301, 5.28186697409185948060013263027, 5.89706123978545200226975534533, 7.33807092747286015892275102407, 8.253158637349628081886891555853, 8.717431507189508441762219730357, 9.474835345348604022768159047682, 10.48578500125122597595345153564

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