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Properties

Label 2-273-91.51-c1-0-6
Degree $2$
Conductor $273$
Sign $0.164 - 0.986i$
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  + (1.69 + 0.977i)2-s + (0.5 + 0.866i)3-s + (0.910 + 1.57i)4-s + (−0.130 − 0.0753i)5-s + 1.95i·6-s + (−0.807 + 2.51i)7-s − 0.348i·8-s + (−0.499 + 0.866i)9-s + (−0.147 − 0.255i)10-s + (0.669 − 0.386i)11-s + (−0.910 + 1.57i)12-s + (3.25 + 1.54i)13-s + (−3.82 + 3.47i)14-s − 0.150i·15-s + (2.16 − 3.74i)16-s + (0.461 + 0.799i)17-s + ⋯
L(s)  = 1  + (1.19 + 0.691i)2-s + (0.288 + 0.499i)3-s + (0.455 + 0.788i)4-s + (−0.0583 − 0.0337i)5-s + 0.798i·6-s + (−0.305 + 0.952i)7-s − 0.123i·8-s + (−0.166 + 0.288i)9-s + (−0.0465 − 0.0807i)10-s + (0.201 − 0.116i)11-s + (−0.262 + 0.455i)12-s + (0.903 + 0.427i)13-s + (−1.02 + 0.929i)14-s − 0.0389i·15-s + (0.540 − 0.936i)16-s + (0.111 + 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82027 + 1.54226i\)
\(L(\frac12)\) \(\approx\) \(1.82027 + 1.54226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.807 - 2.51i)T \)
13 \( 1 + (-3.25 - 1.54i)T \)
good2 \( 1 + (-1.69 - 0.977i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.130 + 0.0753i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.669 + 0.386i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.461 - 0.799i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.54 + 3.77i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.61 + 6.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.68T + 29T^{2} \)
31 \( 1 + (0.532 - 0.307i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.09 + 1.78i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 + 0.868T + 43T^{2} \)
47 \( 1 + (4.89 + 2.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.65 - 8.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.07 + 4.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.90 - 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.56 + 2.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.73iT - 71T^{2} \)
73 \( 1 + (4.19 - 2.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.82 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.57iT - 83T^{2} \)
89 \( 1 + (-8.95 - 5.16i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.78iT - 97T^{2} \)
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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

−12.45539249978606882578066001697, −11.37546446311502588550547549870, −10.25155400549467786547842288338, −9.021474568160026010682811567641, −8.344369789306987692590219172178, −6.67172548292392512124332031774, −6.11459676824005411961571527596, −4.87377442583549813867253269169, −4.01169118016921743657173684492, −2.70378694812219430235124717395, 1.67374415519274240935769788802, 3.34062328139357157913535977945, 3.98760081594187536007418941228, 5.43882926915868701447145409146, 6.53720794593336632372153977880, 7.68627197108185868151744750931, 8.756665303777009879321903661262, 10.18049298074574002853315794219, 11.01552074369849585912578398486, 11.85199695118794940752497954705

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