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Properties

Label 2-273-91.76-c1-0-10
Degree $2$
Conductor $273$
Sign $0.983 + 0.179i$
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  + (0.0578 − 0.215i)2-s + (0.866 − 0.5i)3-s + (1.68 + 0.975i)4-s + (−1.08 + 1.08i)5-s + (−0.0578 − 0.215i)6-s + (−0.643 − 2.56i)7-s + (0.624 − 0.624i)8-s + (0.499 − 0.866i)9-s + (0.171 + 0.297i)10-s + (4.17 + 1.11i)11-s + 1.95·12-s + (3.01 − 1.98i)13-s + (−0.591 − 0.00954i)14-s + (−0.398 + 1.48i)15-s + (1.85 + 3.20i)16-s + (−3.78 + 6.55i)17-s + ⋯
L(s)  = 1  + (0.0408 − 0.152i)2-s + (0.499 − 0.288i)3-s + (0.844 + 0.487i)4-s + (−0.486 + 0.486i)5-s + (−0.0236 − 0.0880i)6-s + (−0.243 − 0.969i)7-s + (0.220 − 0.220i)8-s + (0.166 − 0.288i)9-s + (0.0543 + 0.0941i)10-s + (1.25 + 0.337i)11-s + 0.562·12-s + (0.835 − 0.550i)13-s + (−0.157 − 0.00255i)14-s + (−0.102 + 0.383i)15-s + (0.462 + 0.801i)16-s + (−0.917 + 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.983 + 0.179i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70797 - 0.154505i\)
\(L(\frac12)\) \(\approx\) \(1.70797 - 0.154505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.643 + 2.56i)T \)
13 \( 1 + (-3.01 + 1.98i)T \)
good2 \( 1 + (-0.0578 + 0.215i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.08 - 1.08i)T - 5iT^{2} \)
11 \( 1 + (-4.17 - 1.11i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.78 - 6.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.31 + 4.91i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.85 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.66 + 4.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.09 - 1.09i)T - 31iT^{2} \)
37 \( 1 + (2.82 + 0.757i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.37 + 0.369i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.58 + 3.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.26 + 5.26i)T + 47iT^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + (-8.73 + 2.34i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.1 - 5.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.28 - 4.78i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.25 - 1.40i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.04 + 8.04i)T + 73iT^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 + (1.13 - 1.13i)T - 83iT^{2} \)
89 \( 1 + (-0.217 + 0.811i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.35 - 16.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.71311454917248460118174767471, −11.06241649128153220091362093148, −10.22361894841389841066695661186, −8.818007018769461490120114948268, −7.88076892282574491333482253261, −6.89486453208491451772996837220, −6.41460922860302519455178018541, −3.99948114810695235149319999446, −3.46970718995774529818851395291, −1.77078483274508894328788485856, 1.81437368075205928907968611908, 3.34310137263308699315596602089, 4.71118405889407192686937302616, 6.05190599983470251162072251421, 6.84584564240446001154270465532, 8.258941319741100211149072329131, 9.008590379076185432108430661056, 9.859718142653133441148558347611, 11.31050780906764136128745605366, 11.67463720914502480542539773575

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