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Properties

Label 2-273-91.80-c1-0-9
Degree $2$
Conductor $273$
Sign $0.505 - 0.862i$
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  + (0.323 + 1.20i)2-s + i·3-s + (0.376 − 0.217i)4-s + (0.986 − 3.68i)5-s + (−1.20 + 0.323i)6-s + (−1.80 + 1.93i)7-s + (2.15 + 2.15i)8-s − 9-s + 4.76·10-s + (3.16 + 3.16i)11-s + (0.217 + 0.376i)12-s + (3.57 − 0.437i)13-s + (−2.91 − 1.56i)14-s + (3.68 + 0.986i)15-s + (−1.47 + 2.54i)16-s + (0.601 + 1.04i)17-s + ⋯
L(s)  = 1  + (0.228 + 0.854i)2-s + 0.577i·3-s + (0.188 − 0.108i)4-s + (0.441 − 1.64i)5-s + (−0.493 + 0.132i)6-s + (−0.683 + 0.729i)7-s + (0.761 + 0.761i)8-s − 0.333·9-s + 1.50·10-s + (0.953 + 0.953i)11-s + (0.0627 + 0.108i)12-s + (0.992 − 0.121i)13-s + (−0.780 − 0.417i)14-s + (0.950 + 0.254i)15-s + (−0.367 + 0.637i)16-s + (0.145 + 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46615 + 0.839898i\)
\(L(\frac12)\) \(\approx\) \(1.46615 + 0.839898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (1.80 - 1.93i)T \)
13 \( 1 + (-3.57 + 0.437i)T \)
good2 \( 1 + (-0.323 - 1.20i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.986 + 3.68i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.16 - 3.16i)T + 11iT^{2} \)
17 \( 1 + (-0.601 - 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.79 + 3.79i)T + 19iT^{2} \)
23 \( 1 + (-2.92 - 1.69i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.16 - 1.38i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.75 - 1.80i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.914 + 3.41i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.74 + 5.04i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.98 - 1.33i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.14 - 1.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.88 + 1.57i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.40iT - 61T^{2} \)
67 \( 1 + (3.83 - 3.83i)T - 67iT^{2} \)
71 \( 1 + (-3.05 - 11.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.92 - 7.16i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.76 - 9.98i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.00 + 5.00i)T + 83iT^{2} \)
89 \( 1 + (-1.36 - 5.08i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.69 + 0.722i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.23099857269014905444101175597, −11.17022886831992190584897471091, −9.861563538667646882025073964539, −9.028986790385592448023554873095, −8.459499671971080490870740381335, −6.88240739232469872056634066915, −5.87761560378102540916551345581, −5.15417747750492298197743313101, −4.06982273760053112586011850042, −1.83597789176521163750799918762, 1.66556516671912696655465907290, 3.22247581365414680023612116714, 3.63837648183352152076833811250, 6.19350015011844208101512933083, 6.65293327269404157608758980922, 7.52951522325430348925039668810, 9.082719596577406720092219765463, 10.41285783174270375378829909802, 10.82908708965597949098498249053, 11.53403352001996178805837407265

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