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Properties

Label 2-273-91.83-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.401 - 0.916i$
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  + (−0.786 + 0.786i)2-s − i·3-s + 0.762i·4-s + (2.85 + 2.85i)5-s + (0.786 + 0.786i)6-s + (−1.83 + 1.90i)7-s + (−2.17 − 2.17i)8-s − 9-s − 4.48·10-s + (−1.37 − 1.37i)11-s + 0.762·12-s + (2.79 + 2.27i)13-s + (−0.0482 − 2.94i)14-s + (2.85 − 2.85i)15-s + 1.89·16-s − 3.67·17-s + ⋯
L(s)  = 1  + (−0.556 + 0.556i)2-s − 0.577i·3-s + 0.381i·4-s + (1.27 + 1.27i)5-s + (0.321 + 0.321i)6-s + (−0.695 + 0.718i)7-s + (−0.768 − 0.768i)8-s − 0.333·9-s − 1.41·10-s + (−0.413 − 0.413i)11-s + 0.220·12-s + (0.775 + 0.631i)13-s + (−0.0128 − 0.786i)14-s + (0.736 − 0.736i)15-s + 0.473·16-s − 0.892·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.401 - 0.916i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.401 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534725 + 0.817891i\)
\(L(\frac12)\) \(\approx\) \(0.534725 + 0.817891i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 - 1.90i)T \)
13 \( 1 + (-2.79 - 2.27i)T \)
good2 \( 1 + (0.786 - 0.786i)T - 2iT^{2} \)
5 \( 1 + (-2.85 - 2.85i)T + 5iT^{2} \)
11 \( 1 + (1.37 + 1.37i)T + 11iT^{2} \)
17 \( 1 + 3.67T + 17T^{2} \)
19 \( 1 + (-1.10 - 1.10i)T + 19iT^{2} \)
23 \( 1 + 0.653iT - 23T^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 + (-5.70 - 5.70i)T + 31iT^{2} \)
37 \( 1 + (-3.46 - 3.46i)T + 37iT^{2} \)
41 \( 1 + (-2.67 - 2.67i)T + 41iT^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (-3.84 + 3.84i)T - 47iT^{2} \)
53 \( 1 + 0.919T + 53T^{2} \)
59 \( 1 + (-8.69 + 8.69i)T - 59iT^{2} \)
61 \( 1 - 5.92iT - 61T^{2} \)
67 \( 1 + (8.44 - 8.44i)T - 67iT^{2} \)
71 \( 1 + (-8.55 + 8.55i)T - 71iT^{2} \)
73 \( 1 + (-6.74 + 6.74i)T - 73iT^{2} \)
79 \( 1 - 9.87T + 79T^{2} \)
83 \( 1 + (8.71 + 8.71i)T + 83iT^{2} \)
89 \( 1 + (4.94 - 4.94i)T - 89iT^{2} \)
97 \( 1 + (0.857 + 0.857i)T + 97iT^{2} \)
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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

−12.25549068210475379503707556273, −11.18646209120490038325548286162, −10.11904870088690731339486334687, −9.185502927455163587025983609139, −8.415538357835399395845102321716, −7.01943476460345572995169162064, −6.48847136812036270554329159787, −5.77329751482227061006505191427, −3.32301798039864515144177855881, −2.32752590306520533766534334033, 0.915110704236847474709858923405, 2.49975275448303117950539589522, 4.38959568690115907949155071690, 5.50184312306115021259567470556, 6.29046285787189726341648239455, 8.172745437629383804352071096789, 9.250418247337370281825661495440, 9.637443860936292590063736243346, 10.41517678126851712496883083162, 11.23407757351133869951708197536

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