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Properties

Label 2-273-91.23-c1-0-14
Degree $2$
Conductor $273$
Sign $0.588 + 0.808i$
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  + 2.50i·2-s + (0.5 − 0.866i)3-s − 4.29·4-s + (−2.97 − 1.71i)5-s + (2.17 + 1.25i)6-s + (−0.0473 − 2.64i)7-s − 5.75i·8-s + (−0.499 − 0.866i)9-s + (4.30 − 7.45i)10-s + (−4.35 − 2.51i)11-s + (−2.14 + 3.71i)12-s + (0.135 + 3.60i)13-s + (6.63 − 0.118i)14-s + (−2.97 + 1.71i)15-s + 5.85·16-s + 0.199·17-s + ⋯
L(s)  = 1  + 1.77i·2-s + (0.288 − 0.499i)3-s − 2.14·4-s + (−1.32 − 0.766i)5-s + (0.887 + 0.512i)6-s + (−0.0178 − 0.999i)7-s − 2.03i·8-s + (−0.166 − 0.288i)9-s + (1.36 − 2.35i)10-s + (−1.31 − 0.757i)11-s + (−0.619 + 1.07i)12-s + (0.0375 + 0.999i)13-s + (1.77 − 0.0317i)14-s + (−0.766 + 0.442i)15-s + 1.46·16-s + 0.0483·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.588 + 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408231 - 0.207660i\)
\(L(\frac12)\) \(\approx\) \(0.408231 - 0.207660i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0473 + 2.64i)T \)
13 \( 1 + (-0.135 - 3.60i)T \)
good2 \( 1 - 2.50iT - 2T^{2} \)
5 \( 1 + (2.97 + 1.71i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.35 + 2.51i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.199T + 17T^{2} \)
19 \( 1 + (-3.99 + 2.30i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.52T + 23T^{2} \)
29 \( 1 + (-1.58 - 2.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.65 - 2.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.11iT - 37T^{2} \)
41 \( 1 + (-4.67 + 2.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.27 + 2.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.19 - 3.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.215 + 0.373i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.838iT - 59T^{2} \)
61 \( 1 + (-1.33 - 2.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.67 + 5.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.95 + 2.86i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.73 + 1.57i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.25 + 3.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.7iT - 83T^{2} \)
89 \( 1 - 2.34iT - 89T^{2} \)
97 \( 1 + (5.02 + 2.89i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.04144617951352251442219332463, −10.80757963003768372246712216323, −9.233087012040823880699192799339, −8.423604798791274340161257267157, −7.56205298430277590803103283993, −7.31669318054477136346018347539, −5.87762654670624213683777387912, −4.70640012349487686189823727664, −3.76375727642635654165829230341, −0.33435799610454784557587807216, 2.47270273305782182852028916494, 3.21819443209500588733220011915, 4.28715603034924598534420663597, 5.49532121147472481058779118999, 7.73292755293299564617660781580, 8.302072152756266354182539637599, 9.723710645363862989903011353341, 10.26208910791977133833216838102, 11.14137908145793808353405864343, 11.94612760793829641761624897815

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