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Properties

Label 2-273-39.5-c1-0-22
Degree $2$
Conductor $273$
Sign $0.894 + 0.446i$
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  + (0.636 + 0.636i)2-s + (1.71 − 0.215i)3-s − 1.18i·4-s + (−1.55 − 1.55i)5-s + (1.23 + 0.957i)6-s + (−0.707 − 0.707i)7-s + (2.03 − 2.03i)8-s + (2.90 − 0.740i)9-s − 1.97i·10-s + (−2.77 + 2.77i)11-s + (−0.256 − 2.04i)12-s + (3.58 + 0.345i)13-s − 0.900i·14-s + (−3.00 − 2.33i)15-s + 0.209·16-s + 6.44·17-s + ⋯
L(s)  = 1  + (0.450 + 0.450i)2-s + (0.992 − 0.124i)3-s − 0.594i·4-s + (−0.694 − 0.694i)5-s + (0.502 + 0.390i)6-s + (−0.267 − 0.267i)7-s + (0.718 − 0.718i)8-s + (0.969 − 0.246i)9-s − 0.625i·10-s + (−0.835 + 0.835i)11-s + (−0.0739 − 0.589i)12-s + (0.995 + 0.0958i)13-s − 0.240i·14-s + (−0.775 − 0.602i)15-s + 0.0523·16-s + 1.56·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89870 - 0.447265i\)
\(L(\frac12)\) \(\approx\) \(1.89870 - 0.447265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.215i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-3.58 - 0.345i)T \)
good2 \( 1 + (-0.636 - 0.636i)T + 2iT^{2} \)
5 \( 1 + (1.55 + 1.55i)T + 5iT^{2} \)
11 \( 1 + (2.77 - 2.77i)T - 11iT^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + (3.44 - 3.44i)T - 19iT^{2} \)
23 \( 1 + 4.40T + 23T^{2} \)
29 \( 1 - 9.71iT - 29T^{2} \)
31 \( 1 + (4.02 - 4.02i)T - 31iT^{2} \)
37 \( 1 + (0.492 + 0.492i)T + 37iT^{2} \)
41 \( 1 + (-1.74 - 1.74i)T + 41iT^{2} \)
43 \( 1 - 4.10iT - 43T^{2} \)
47 \( 1 + (-8.82 + 8.82i)T - 47iT^{2} \)
53 \( 1 - 0.952iT - 53T^{2} \)
59 \( 1 + (-3.38 + 3.38i)T - 59iT^{2} \)
61 \( 1 + 7.73T + 61T^{2} \)
67 \( 1 + (4.28 - 4.28i)T - 67iT^{2} \)
71 \( 1 + (9.09 + 9.09i)T + 71iT^{2} \)
73 \( 1 + (-5.75 - 5.75i)T + 73iT^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + (-2.69 - 2.69i)T + 83iT^{2} \)
89 \( 1 + (1.20 - 1.20i)T - 89iT^{2} \)
97 \( 1 + (1.06 - 1.06i)T - 97iT^{2} \)
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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

−12.36579754126632512780791662232, −10.57100116992201484676834630039, −9.990893689332669699756664791693, −8.780211688422876885027502983591, −7.87444814473262959633574093107, −7.05696326020907303016442896626, −5.71096235600603730829959758259, −4.49395597298792999233516894136, −3.56582238865999630395103889056, −1.50497482395294389033666785689, 2.51368370841109339130645541866, 3.38235258773672920134989961356, 4.13902256847300355004253769541, 5.87504807300241916977986619506, 7.52133284339158039707075186413, 7.977236666538070553034765020478, 8.927806751825342080273763414468, 10.30789906606549399164325708335, 11.08850480228565342521132589821, 12.01501562813305283064798797553

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