Properties

Label 2-273-273.272-c1-0-8
Degree $2$
Conductor $273$
Sign $0.131 - 0.991i$
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  − 1.77·2-s + (1.65 + 0.525i)3-s + 1.15·4-s + 3.85i·5-s + (−2.93 − 0.933i)6-s + (1.56 − 2.12i)7-s + 1.50·8-s + (2.44 + 1.73i)9-s − 6.84i·10-s − 2.15·11-s + (1.90 + 0.606i)12-s + (−2.48 + 2.61i)13-s + (−2.78 + 3.78i)14-s + (−2.02 + 6.36i)15-s − 4.97·16-s − 1.53·17-s + ⋯
L(s)  = 1  − 1.25·2-s + (0.952 + 0.303i)3-s + 0.576·4-s + 1.72i·5-s + (−1.19 − 0.381i)6-s + (0.593 − 0.804i)7-s + 0.531·8-s + (0.815 + 0.578i)9-s − 2.16i·10-s − 0.649·11-s + (0.549 + 0.175i)12-s + (−0.688 + 0.725i)13-s + (−0.745 + 1.01i)14-s + (−0.523 + 1.64i)15-s − 1.24·16-s − 0.371·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.131 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678515 + 0.594638i\)
\(L(\frac12)\) \(\approx\) \(0.678515 + 0.594638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 0.525i)T \)
7 \( 1 + (-1.56 + 2.12i)T \)
13 \( 1 + (2.48 - 2.61i)T \)
good2 \( 1 + 1.77T + 2T^{2} \)
5 \( 1 - 3.85iT - 5T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
23 \( 1 - 0.929iT - 23T^{2} \)
29 \( 1 - 2.00iT - 29T^{2} \)
31 \( 1 - 0.380T + 31T^{2} \)
37 \( 1 - 8.77iT - 37T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 - 5.97T + 43T^{2} \)
47 \( 1 + 1.59iT - 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 - 0.317iT - 59T^{2} \)
61 \( 1 + 1.21iT - 61T^{2} \)
67 \( 1 + 11.0iT - 67T^{2} \)
71 \( 1 + 7.83T + 71T^{2} \)
73 \( 1 + 1.37T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 7.59iT - 83T^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−11.55079262430021101537918183022, −10.73986420019936825792039901729, −10.10824696222412465201104799778, −9.463312661284577955846887848359, −8.175908020936996883405265125851, −7.41146865882512327437384708247, −6.95314068916535005626114010927, −4.71743451219271011031313990858, −3.30828940981271904074618320024, −2.01879505182641099005345962843, 1.01938754037419571432028315885, 2.37804995497929602958442361814, 4.50100749402729524664434238835, 5.48206696650387404349878862511, 7.64972948584783602877908904261, 7.911676791902143007863035260627, 9.013921938906172086754891343988, 9.207713188838001273137504700917, 10.30222261668978089576768932606, 11.77364627973191367535555941656

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