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Properties

Label 2-273-39.8-c1-0-5
Degree $2$
Conductor $273$
Sign $0.411 - 0.911i$
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  + (1.34 − 1.34i)2-s + (−0.469 + 1.66i)3-s − 1.63i·4-s + (−3.06 + 3.06i)5-s + (1.61 + 2.87i)6-s + (−0.707 + 0.707i)7-s + (0.496 + 0.496i)8-s + (−2.55 − 1.56i)9-s + 8.26i·10-s + (1.91 + 1.91i)11-s + (2.71 + 0.766i)12-s + (−2.89 − 2.15i)13-s + 1.90i·14-s + (−3.67 − 6.55i)15-s + 4.60·16-s + 5.21·17-s + ⋯
L(s)  = 1  + (0.952 − 0.952i)2-s + (−0.271 + 0.962i)3-s − 0.815i·4-s + (−1.37 + 1.37i)5-s + (0.658 + 1.17i)6-s + (−0.267 + 0.267i)7-s + (0.175 + 0.175i)8-s + (−0.852 − 0.521i)9-s + 2.61i·10-s + (0.577 + 0.577i)11-s + (0.785 + 0.221i)12-s + (−0.801 − 0.597i)13-s + 0.509i·14-s + (−0.948 − 1.69i)15-s + 1.15·16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.411 - 0.911i$
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: \((2,\ 273,\ (\ :1/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17837 + 0.761097i\)
\(L(\frac12)\) \(\approx\) \(1.17837 + 0.761097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.469 - 1.66i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (2.89 + 2.15i)T \)
good2 \( 1 + (-1.34 + 1.34i)T - 2iT^{2} \)
5 \( 1 + (3.06 - 3.06i)T - 5iT^{2} \)
11 \( 1 + (-1.91 - 1.91i)T + 11iT^{2} \)
17 \( 1 - 5.21T + 17T^{2} \)
19 \( 1 + (-2.19 - 2.19i)T + 19iT^{2} \)
23 \( 1 + 1.85T + 23T^{2} \)
29 \( 1 - 5.69iT - 29T^{2} \)
31 \( 1 + (-2.18 - 2.18i)T + 31iT^{2} \)
37 \( 1 + (-1.32 + 1.32i)T - 37iT^{2} \)
41 \( 1 + (2.07 - 2.07i)T - 41iT^{2} \)
43 \( 1 + 3.43iT - 43T^{2} \)
47 \( 1 + (-0.505 - 0.505i)T + 47iT^{2} \)
53 \( 1 - 7.99iT - 53T^{2} \)
59 \( 1 + (2.49 + 2.49i)T + 59iT^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 + (-8.77 - 8.77i)T + 67iT^{2} \)
71 \( 1 + (0.255 - 0.255i)T - 71iT^{2} \)
73 \( 1 + (-3.52 + 3.52i)T - 73iT^{2} \)
79 \( 1 + 7.05T + 79T^{2} \)
83 \( 1 + (-4.05 + 4.05i)T - 83iT^{2} \)
89 \( 1 + (12.8 + 12.8i)T + 89iT^{2} \)
97 \( 1 + (-4.00 - 4.00i)T + 97iT^{2} \)
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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

−12.09636626377893479887258289816, −11.29573168811349869053648192219, −10.42637768199811439033641653728, −9.871049153042716060198177921829, −8.134129412557443267035346214205, −7.11442528146000850027951881637, −5.64569803917051428504598980745, −4.44157905718348988894226626706, −3.50195387208219229193554085452, −2.91242171830351878187332781590, 0.883398936310574813865663018643, 3.64448034457495321955092638286, 4.71206444919670276636477849564, 5.61122066665515461532487250506, 6.78575960751178443583055275453, 7.66711830944874478454136205405, 8.247178976993735290152455938375, 9.598943535589869392024214782787, 11.45428649617066881473802216157, 12.04172162462298846923044852463

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