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Properties

Label 2-272-16.5-c1-0-21
Degree $2$
Conductor $272$
Sign $0.570 + 0.820i$
Analytic cond. $2.17193$
Root an. cond. $1.47374$
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  + (−1.13 + 0.847i)2-s + (0.907 − 0.907i)3-s + (0.562 − 1.91i)4-s + (0.218 + 0.218i)5-s + (−0.258 + 1.79i)6-s − 4.34i·7-s + (0.989 + 2.64i)8-s + 1.35i·9-s + (−0.432 − 0.0620i)10-s + (−3.20 − 3.20i)11-s + (−1.23 − 2.25i)12-s + (1.70 − 1.70i)13-s + (3.68 + 4.92i)14-s + 0.396·15-s + (−3.36 − 2.16i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.800 + 0.599i)2-s + (0.524 − 0.524i)3-s + (0.281 − 0.959i)4-s + (0.0976 + 0.0976i)5-s + (−0.105 + 0.733i)6-s − 1.64i·7-s + (0.349 + 0.936i)8-s + 0.450i·9-s + (−0.136 − 0.0196i)10-s + (−0.964 − 0.964i)11-s + (−0.355 − 0.650i)12-s + (0.472 − 0.472i)13-s + (0.985 + 1.31i)14-s + 0.102·15-s + (−0.841 − 0.540i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.570 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.570 + 0.820i$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ 0.570 + 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859042 - 0.448953i\)
\(L(\frac12)\) \(\approx\) \(0.859042 - 0.448953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.847i)T \)
17 \( 1 + T \)
good3 \( 1 + (-0.907 + 0.907i)T - 3iT^{2} \)
5 \( 1 + (-0.218 - 0.218i)T + 5iT^{2} \)
7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 + (3.20 + 3.20i)T + 11iT^{2} \)
13 \( 1 + (-1.70 + 1.70i)T - 13iT^{2} \)
19 \( 1 + (-0.630 + 0.630i)T - 19iT^{2} \)
23 \( 1 - 0.799iT - 23T^{2} \)
29 \( 1 + (-2.21 + 2.21i)T - 29iT^{2} \)
31 \( 1 - 6.06T + 31T^{2} \)
37 \( 1 + (-4.96 - 4.96i)T + 37iT^{2} \)
41 \( 1 - 11.0iT - 41T^{2} \)
43 \( 1 + (4.85 + 4.85i)T + 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (3.33 + 3.33i)T + 53iT^{2} \)
59 \( 1 + (3.82 + 3.82i)T + 59iT^{2} \)
61 \( 1 + (3.16 - 3.16i)T - 61iT^{2} \)
67 \( 1 + (7.94 - 7.94i)T - 67iT^{2} \)
71 \( 1 - 6.71iT - 71T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 - 1.19T + 79T^{2} \)
83 \( 1 + (0.310 - 0.310i)T - 83iT^{2} \)
89 \( 1 + 0.417iT - 89T^{2} \)
97 \( 1 - 1.28T + 97T^{2} \)
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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

−11.35624967633013746513471767354, −10.53933398752192794038739492659, −10.01625166107773980444582116333, −8.399145211593633301848686641836, −8.007370192020288575199319554845, −7.11323636591863712061468780347, −6.08688616981393088408261208055, −4.65823691857091697458996006137, −2.80088620144029176936178471782, −0.952191651848680683715234603067, 2.11973883993109244704310768409, 3.14120790253735772793151543569, 4.62099360140903959931265726268, 6.11275443953265911906938758228, 7.51811720742024510874207198594, 8.692284528017657093384405200159, 9.135981172993474714714621720128, 9.938498154211998931250189439112, 10.97744655290974712027024410135, 12.15060038965875995584594381357

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