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Properties

Label 2-273-91.31-c1-0-7
Degree $2$
Conductor $273$
Sign $0.994 - 0.105i$
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  + (−2.62 − 0.703i)2-s + (0.866 + 0.5i)3-s + (4.65 + 2.69i)4-s + (−0.276 + 1.03i)5-s + (−1.92 − 1.92i)6-s + (2.15 − 1.53i)7-s + (−6.49 − 6.49i)8-s + (0.499 + 0.866i)9-s + (1.45 − 2.51i)10-s + (4.19 − 1.12i)11-s + (2.69 + 4.65i)12-s + (−2.90 + 2.13i)13-s + (−6.73 + 2.51i)14-s + (−0.756 + 0.756i)15-s + (7.09 + 12.2i)16-s + (−2.51 + 4.35i)17-s + ⋯
L(s)  = 1  + (−1.85 − 0.497i)2-s + (0.499 + 0.288i)3-s + (2.32 + 1.34i)4-s + (−0.123 + 0.462i)5-s + (−0.784 − 0.784i)6-s + (0.813 − 0.581i)7-s + (−2.29 − 2.29i)8-s + (0.166 + 0.288i)9-s + (0.459 − 0.796i)10-s + (1.26 − 0.338i)11-s + (0.776 + 1.34i)12-s + (−0.804 + 0.593i)13-s + (−1.79 + 0.673i)14-s + (−0.195 + 0.195i)15-s + (1.77 + 3.07i)16-s + (−0.610 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.994 - 0.105i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.994 - 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.733598 + 0.0386677i\)
\(L(\frac12)\) \(\approx\) \(0.733598 + 0.0386677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2.15 + 1.53i)T \)
13 \( 1 + (2.90 - 2.13i)T \)
good2 \( 1 + (2.62 + 0.703i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.276 - 1.03i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.19 + 1.12i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.51 - 4.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.861 + 3.21i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.25 + 1.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.78T + 29T^{2} \)
31 \( 1 + (0.585 - 0.156i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.64 - 6.15i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.51 + 2.51i)T + 41iT^{2} \)
43 \( 1 + 5.67iT - 43T^{2} \)
47 \( 1 + (-7.61 - 2.04i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.28 - 2.21i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.142 - 0.531i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.16 - 3.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.79 - 6.69i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.29 - 6.29i)T - 71iT^{2} \)
73 \( 1 + (1.13 + 4.24i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.13 - 7.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.35 + 6.35i)T + 83iT^{2} \)
89 \( 1 + (3.48 + 0.934i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.0 + 10.0i)T + 97iT^{2} \)
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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

−11.45063202359954101187071365178, −10.76346346781789394737858007519, −10.02900403181961947880165376513, −8.904192339966960626742089812904, −8.508224701766035967026681687809, −7.23768055252543632301795038946, −6.70788353214355757027418821454, −4.23621805704986455339520047484, −2.83148380330601736344667538198, −1.42662189598749130851247662143, 1.18877344385990801718671771405, 2.52118973987776218195815217952, 4.99035339148591122619700595095, 6.42171807827395807075283411289, 7.36265107150191587511001161202, 8.173190996537267967565735319461, 8.978591296307688673180961959983, 9.516379159791341316225387768606, 10.65283654832916329169873757214, 11.78028318984349810666977957968

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