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Properties

Label 2-273-91.81-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.853 - 0.521i$
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  + (−0.904 + 1.56i)2-s + 3-s + (−0.637 − 1.10i)4-s + (1.98 + 3.44i)5-s + (−0.904 + 1.56i)6-s + (−2.60 − 0.469i)7-s − 1.31·8-s + 9-s − 7.19·10-s − 0.286·11-s + (−0.637 − 1.10i)12-s + (3.60 + 0.185i)13-s + (3.09 − 3.65i)14-s + (1.98 + 3.44i)15-s + (2.46 − 4.26i)16-s + (2.30 + 3.98i)17-s + ⋯
L(s)  = 1  + (−0.639 + 1.10i)2-s + 0.577·3-s + (−0.318 − 0.552i)4-s + (0.888 + 1.53i)5-s + (−0.369 + 0.639i)6-s + (−0.984 − 0.177i)7-s − 0.463·8-s + 0.333·9-s − 2.27·10-s − 0.0864·11-s + (−0.184 − 0.318i)12-s + (0.998 + 0.0515i)13-s + (0.826 − 0.977i)14-s + (0.513 + 0.888i)15-s + (0.615 − 1.06i)16-s + (0.558 + 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.853 - 0.521i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.853 - 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303708 + 1.07929i\)
\(L(\frac12)\) \(\approx\) \(0.303708 + 1.07929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + (2.60 + 0.469i)T \)
13 \( 1 + (-3.60 - 0.185i)T \)
good2 \( 1 + (0.904 - 1.56i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.98 - 3.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.286T + 11T^{2} \)
17 \( 1 + (-2.30 - 3.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 6.96T + 19T^{2} \)
23 \( 1 + (-3.61 + 6.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.421 - 0.730i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.212 - 0.368i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.18 + 3.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.509 + 0.883i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.585 + 1.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.71 - 4.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.574 - 0.994i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.42 - 4.20i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.16T + 61T^{2} \)
67 \( 1 - 1.57T + 67T^{2} \)
71 \( 1 + (3.22 - 5.58i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.24 + 14.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.84 + 6.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + (-1.10 + 1.91i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.52 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.59290996067308600936898122689, −10.77078811342810311005725396876, −10.30204641147968580130994481198, −9.241463319072423904503410289439, −8.426103932109820207551721133873, −7.24364135626803046805759226925, −6.41370691213343811536073547530, −6.05653400674286014116622402438, −3.63682710492485600692451889561, −2.54301924651831168008027543092, 1.05248843105080146216801177475, 2.36585021781316013412817456872, 3.71616821977742475232935147710, 5.33991703921036333504216924492, 6.41867460280339732063122999868, 8.271561833809006133221653343319, 9.013634775298500826107315558832, 9.528407627288371431330386014286, 10.25953171253866997041731067779, 11.49969464089490348048265194384

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