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Properties

Label 2-273-39.8-c1-0-10
Degree $2$
Conductor $273$
Sign $0.959 - 0.282i$
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  + (−0.946 + 0.946i)2-s + (−1.73 − 0.0399i)3-s + 0.206i·4-s + (1.18 − 1.18i)5-s + (1.67 − 1.60i)6-s + (0.707 − 0.707i)7-s + (−2.08 − 2.08i)8-s + (2.99 + 0.138i)9-s + 2.25i·10-s + (−3.22 − 3.22i)11-s + (0.00825 − 0.357i)12-s + (3.54 + 0.652i)13-s + 1.33i·14-s + (−2.10 + 2.01i)15-s + 3.54·16-s + 4.20·17-s + ⋯
L(s)  = 1  + (−0.669 + 0.669i)2-s + (−0.999 − 0.0230i)3-s + 0.103i·4-s + (0.531 − 0.531i)5-s + (0.684 − 0.653i)6-s + (0.267 − 0.267i)7-s + (−0.738 − 0.738i)8-s + (0.998 + 0.0461i)9-s + 0.711i·10-s + (−0.973 − 0.973i)11-s + (0.00238 − 0.103i)12-s + (0.983 + 0.180i)13-s + 0.357i·14-s + (−0.543 + 0.518i)15-s + 0.886·16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.959 - 0.282i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.959 - 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718468 + 0.103564i\)
\(L(\frac12)\) \(\approx\) \(0.718468 + 0.103564i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0399i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-3.54 - 0.652i)T \)
good2 \( 1 + (0.946 - 0.946i)T - 2iT^{2} \)
5 \( 1 + (-1.18 + 1.18i)T - 5iT^{2} \)
11 \( 1 + (3.22 + 3.22i)T + 11iT^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 + (-1.77 - 1.77i)T + 19iT^{2} \)
23 \( 1 - 7.43T + 23T^{2} \)
29 \( 1 + 0.211iT - 29T^{2} \)
31 \( 1 + (1.20 + 1.20i)T + 31iT^{2} \)
37 \( 1 + (-6.89 + 6.89i)T - 37iT^{2} \)
41 \( 1 + (6.06 - 6.06i)T - 41iT^{2} \)
43 \( 1 + 8.80iT - 43T^{2} \)
47 \( 1 + (3.55 + 3.55i)T + 47iT^{2} \)
53 \( 1 + 5.75iT - 53T^{2} \)
59 \( 1 + (-7.52 - 7.52i)T + 59iT^{2} \)
61 \( 1 - 6.83T + 61T^{2} \)
67 \( 1 + (6.97 + 6.97i)T + 67iT^{2} \)
71 \( 1 + (1.27 - 1.27i)T - 71iT^{2} \)
73 \( 1 + (3.21 - 3.21i)T - 73iT^{2} \)
79 \( 1 - 1.31T + 79T^{2} \)
83 \( 1 + (-7.18 + 7.18i)T - 83iT^{2} \)
89 \( 1 + (5.86 + 5.86i)T + 89iT^{2} \)
97 \( 1 + (-1.14 - 1.14i)T + 97iT^{2} \)
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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

−11.82764829076169598486675134233, −10.94625285635599651391842403239, −9.988173389197522368543827982528, −8.962623782685349806017717609617, −8.020325022323293360442456915561, −7.09228883155304710334974245144, −5.91370981771117531916006957564, −5.25621815152492339990389975445, −3.54141236725061524298656815526, −0.968979450074654715349895035662, 1.30784544176174718942120617538, 2.81692586737756337352864894825, 4.93922295762057372949712447315, 5.73340347170512378312745358767, 6.81846412183087202899898638068, 8.097332989921766722528018374794, 9.469402150765425149019100643640, 10.15016789659731417420263614820, 10.82604354181180930073384585937, 11.51191482271652469521625151383

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