Properties

Label 2-273-91.33-c1-0-10
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: $\chi_{273} (124, \cdot )$
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

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

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