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Properties

Label 2-273-39.5-c1-0-5
Degree $2$
Conductor $273$
Sign $0.262 + 0.965i$
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  + (−1.72 − 1.72i)2-s + (0.578 + 1.63i)3-s + 3.92i·4-s + (−1.26 − 1.26i)5-s + (1.81 − 3.80i)6-s + (−0.707 − 0.707i)7-s + (3.31 − 3.31i)8-s + (−2.33 + 1.88i)9-s + 4.35i·10-s + (2.29 − 2.29i)11-s + (−6.41 + 2.27i)12-s + (3.29 − 1.45i)13-s + 2.43i·14-s + (1.33 − 2.79i)15-s − 3.57·16-s + 3.53·17-s + ⋯
L(s)  = 1  + (−1.21 − 1.21i)2-s + (0.333 + 0.942i)3-s + 1.96i·4-s + (−0.566 − 0.566i)5-s + (0.741 − 1.55i)6-s + (−0.267 − 0.267i)7-s + (1.17 − 1.17i)8-s + (−0.777 + 0.629i)9-s + 1.37i·10-s + (0.692 − 0.692i)11-s + (−1.85 + 0.655i)12-s + (0.914 − 0.404i)13-s + 0.650i·14-s + (0.344 − 0.722i)15-s − 0.892·16-s + 0.858·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.262 + 0.965i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.262 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554138 - 0.423674i\)
\(L(\frac12)\) \(\approx\) \(0.554138 - 0.423674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.578 - 1.63i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-3.29 + 1.45i)T \)
good2 \( 1 + (1.72 + 1.72i)T + 2iT^{2} \)
5 \( 1 + (1.26 + 1.26i)T + 5iT^{2} \)
11 \( 1 + (-2.29 + 2.29i)T - 11iT^{2} \)
17 \( 1 - 3.53T + 17T^{2} \)
19 \( 1 + (-4.41 + 4.41i)T - 19iT^{2} \)
23 \( 1 - 6.55T + 23T^{2} \)
29 \( 1 - 2.95iT - 29T^{2} \)
31 \( 1 + (6.35 - 6.35i)T - 31iT^{2} \)
37 \( 1 + (-0.187 - 0.187i)T + 37iT^{2} \)
41 \( 1 + (1.02 + 1.02i)T + 41iT^{2} \)
43 \( 1 + 8.45iT - 43T^{2} \)
47 \( 1 + (-1.07 + 1.07i)T - 47iT^{2} \)
53 \( 1 + 5.36iT - 53T^{2} \)
59 \( 1 + (5.63 - 5.63i)T - 59iT^{2} \)
61 \( 1 - 9.49T + 61T^{2} \)
67 \( 1 + (1.60 - 1.60i)T - 67iT^{2} \)
71 \( 1 + (2.13 + 2.13i)T + 71iT^{2} \)
73 \( 1 + (-8.00 - 8.00i)T + 73iT^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (-4.14 - 4.14i)T + 83iT^{2} \)
89 \( 1 + (1.27 - 1.27i)T - 89iT^{2} \)
97 \( 1 + (-8.78 + 8.78i)T - 97iT^{2} \)
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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

−11.30253795314035262252320470851, −10.77183777389207051709762786388, −9.808482645955295967591895373725, −8.815507828504095743655277203411, −8.612466807465743615867367765850, −7.30262121674506865332717074805, −5.29295297586266671388236250091, −3.73233213231778822908901730515, −3.12354143768807526960321092150, −0.940681809218319769489539282807, 1.35539363486863189877749936138, 3.45614892644281816954370494936, 5.70424362117573158523942628221, 6.55748831591357215881638257178, 7.40134096917951052477314146982, 7.927392926790104965274658127162, 9.080686940576840009899569476434, 9.657297230595173528366788473616, 11.08812831357690971264777279769, 11.97032911466457677326729971301

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