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Properties

Label 2-273-39.5-c1-0-25
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: $\chi_{273} (239, \cdot )$
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

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

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