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Properties

Label 2-273-91.54-c1-0-10
Degree $2$
Conductor $273$
Sign $0.954 + 0.297i$
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  + (0.698 − 0.698i)2-s + (−0.866 − 0.5i)3-s + 1.02i·4-s + (−0.912 + 0.244i)5-s + (−0.954 + 0.255i)6-s + (2.61 − 0.407i)7-s + (2.11 + 2.11i)8-s + (0.499 + 0.866i)9-s + (−0.466 + 0.808i)10-s + (6.35 − 1.70i)11-s + (0.511 − 0.886i)12-s + (−3.43 − 1.08i)13-s + (1.54 − 2.11i)14-s + (0.912 + 0.244i)15-s + 0.904·16-s + 3.73·17-s + ⋯
L(s)  = 1  + (0.494 − 0.494i)2-s + (−0.499 − 0.288i)3-s + 0.511i·4-s + (−0.407 + 0.109i)5-s + (−0.389 + 0.104i)6-s + (0.988 − 0.154i)7-s + (0.746 + 0.746i)8-s + (0.166 + 0.288i)9-s + (−0.147 + 0.255i)10-s + (1.91 − 0.513i)11-s + (0.147 − 0.255i)12-s + (−0.953 − 0.299i)13-s + (0.411 − 0.564i)14-s + (0.235 + 0.0631i)15-s + 0.226·16-s + 0.905·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.954 + 0.297i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.954 + 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52572 - 0.232421i\)
\(L(\frac12)\) \(\approx\) \(1.52572 - 0.232421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.61 + 0.407i)T \)
13 \( 1 + (3.43 + 1.08i)T \)
good2 \( 1 + (-0.698 + 0.698i)T - 2iT^{2} \)
5 \( 1 + (0.912 - 0.244i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-6.35 + 1.70i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 3.73T + 17T^{2} \)
19 \( 1 + (0.325 - 1.21i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.233iT - 23T^{2} \)
29 \( 1 + (2.32 + 4.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.96 - 7.35i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.55 - 3.55i)T + 37iT^{2} \)
41 \( 1 + (-2.49 + 9.29i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (10.4 + 6.00i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.563 + 2.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.04 - 3.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.27 - 5.27i)T - 59iT^{2} \)
61 \( 1 + (2.12 - 1.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.63 + 9.82i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.433 - 1.61i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (7.85 + 2.10i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.942 - 1.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.95 + 9.95i)T + 83iT^{2} \)
89 \( 1 + (2.88 - 2.88i)T - 89iT^{2} \)
97 \( 1 + (-2.41 + 0.647i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.88654496567366853877283318360, −11.38378458508591198849183452284, −10.31543928383248156445680293753, −8.866051164917236777009350410613, −7.82685452615518609800168315218, −7.07537231752671803268477662367, −5.59354757038725434519001973465, −4.42519186384533143115776125992, −3.46958324820745966253290228283, −1.64976258852079618158410389746, 1.49687347090330983089087899183, 4.08229387785380478970170786741, 4.72769667699836670024834449945, 5.82804731691515591065826744272, 6.84425360410831103233750718418, 7.79624343448456255813279259209, 9.318189046680516783345242499034, 9.939363717969403452955961742289, 11.34420286744041546568779709231, 11.73964398589611202327121173526

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