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Properties

Label 2-273-13.12-c1-0-3
Degree $2$
Conductor $273$
Sign $0.342 - 0.939i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
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  + 0.233i·2-s − 3-s + 1.94·4-s + 2.94i·5-s − 0.233i·6-s − i·7-s + 0.921i·8-s + 9-s − 0.687·10-s + 3.62i·11-s − 1.94·12-s + (−3.38 − 1.23i)13-s + 0.233·14-s − 2.94i·15-s + 3.67·16-s + 1.53·17-s + ⋯
L(s)  = 1  + 0.165i·2-s − 0.577·3-s + 0.972·4-s + 1.31i·5-s − 0.0953i·6-s − 0.377i·7-s + 0.325i·8-s + 0.333·9-s − 0.217·10-s + 1.09i·11-s − 0.561·12-s + (−0.939 − 0.342i)13-s + 0.0623·14-s − 0.760i·15-s + 0.918·16-s + 0.371·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04606 + 0.732398i\)
\(L(\frac12)\) \(\approx\) \(1.04606 + 0.732398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + iT \)
13 \( 1 + (3.38 + 1.23i)T \)
good2 \( 1 - 0.233iT - 2T^{2} \)
5 \( 1 - 2.94iT - 5T^{2} \)
11 \( 1 - 3.62iT - 11T^{2} \)
17 \( 1 - 1.53T + 17T^{2} \)
19 \( 1 - 4.10iT - 19T^{2} \)
23 \( 1 - 7.03T + 23T^{2} \)
29 \( 1 + 3.79T + 29T^{2} \)
31 \( 1 - 1.79iT - 31T^{2} \)
37 \( 1 + 7.24iT - 37T^{2} \)
41 \( 1 + 9.15iT - 41T^{2} \)
43 \( 1 - 5.79T + 43T^{2} \)
47 \( 1 + 7.25iT - 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 4.84iT - 59T^{2} \)
61 \( 1 - 2.77T + 61T^{2} \)
67 \( 1 + 14.8iT - 67T^{2} \)
71 \( 1 + 3.62iT - 71T^{2} \)
73 \( 1 + 14.5iT - 73T^{2} \)
79 \( 1 - 6.56T + 79T^{2} \)
83 \( 1 - 7.25iT - 83T^{2} \)
89 \( 1 - 0.636iT - 89T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−12.06606073871970103863441831488, −10.85622517881924753551020875685, −10.58984709492269764374253739628, −9.584338727379922091510486399410, −7.51893974480926736727654416702, −7.27803106750085300850772100744, −6.32028790172054820452278063657, −5.15731459349275679588299606724, −3.44824586229474762633637821250, −2.12247920713668402735662001322, 1.13444297621586274594223265677, 2.89364023055856544602029483310, 4.69741478460604261659142356811, 5.58414330373274965584556389889, 6.65192502676089845684662448688, 7.80809845386472579215069953803, 8.933432722629424654857146771945, 9.816896536178062028906560560627, 11.21323758108477599505358190533, 11.52202572924416195198343244874

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