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Properties

Label 2-273-273.272-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.142 - 0.989i$
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  − 2.35·2-s + (0.813 + 1.52i)3-s + 3.52·4-s + 0.514i·5-s + (−1.91 − 3.59i)6-s + (1.49 + 2.18i)7-s − 3.58·8-s + (−1.67 + 2.48i)9-s − 1.21i·10-s + 2.08·11-s + (2.86 + 5.39i)12-s + (2.92 − 2.11i)13-s + (−3.50 − 5.13i)14-s + (−0.787 + 0.418i)15-s + 1.38·16-s − 0.359·17-s + ⋯
L(s)  = 1  − 1.66·2-s + (0.469 + 0.882i)3-s + 1.76·4-s + 0.230i·5-s + (−0.780 − 1.46i)6-s + (0.563 + 0.826i)7-s − 1.26·8-s + (−0.558 + 0.829i)9-s − 0.382i·10-s + 0.629·11-s + (0.828 + 1.55i)12-s + (0.809 − 0.586i)13-s + (−0.936 − 1.37i)14-s + (−0.203 + 0.108i)15-s + 0.345·16-s − 0.0871·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.142 - 0.989i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465933 + 0.538074i\)
\(L(\frac12)\) \(\approx\) \(0.465933 + 0.538074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.813 - 1.52i)T \)
7 \( 1 + (-1.49 - 2.18i)T \)
13 \( 1 + (-2.92 + 2.11i)T \)
good2 \( 1 + 2.35T + 2T^{2} \)
5 \( 1 - 0.514iT - 5T^{2} \)
11 \( 1 - 2.08T + 11T^{2} \)
17 \( 1 + 0.359T + 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 - 1.93iT - 23T^{2} \)
29 \( 1 - 8.77iT - 29T^{2} \)
31 \( 1 + 7.37T + 31T^{2} \)
37 \( 1 - 2.76iT - 37T^{2} \)
41 \( 1 - 5.37iT - 41T^{2} \)
43 \( 1 + 0.383T + 43T^{2} \)
47 \( 1 + 6.90iT - 47T^{2} \)
53 \( 1 + 9.21iT - 53T^{2} \)
59 \( 1 - 1.01iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 3.71iT - 67T^{2} \)
71 \( 1 - 2.79T + 71T^{2} \)
73 \( 1 - 6.83T + 73T^{2} \)
79 \( 1 - 9.78T + 79T^{2} \)
83 \( 1 + 14.8iT - 83T^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 - 8.34T + 97T^{2} \)
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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

−11.50069067826785076291640509829, −10.91831846505538638707306142920, −10.16145477762794923667039941386, −8.988619081915278239309035571025, −8.747418994500192602876868356406, −7.79828488228926024222237777003, −6.49437936812734623330302067416, −5.08284782083669894948134237015, −3.34696726542988459202992571306, −1.86281979583928556345125295899, 0.949954814132576869331036445555, 2.09312615879258529597638275376, 4.06574322438009314649326940878, 6.27376078618041366621758331377, 7.09172672312021323012001111817, 7.947119370208253491912357616980, 8.738319969108255824786274988153, 9.370726181113866659114167589283, 10.69725550802672511634467964494, 11.29917965091560504498376478999

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