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Properties

Label 2-273-91.83-c1-0-11
Degree $2$
Conductor $273$
Sign $0.995 - 0.0934i$
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  + (1.22 − 1.22i)2-s + i·3-s − 0.999i·4-s + (2 + 2i)5-s + (1.22 + 1.22i)6-s + (−2.44 + i)7-s + (1.22 + 1.22i)8-s − 9-s + 4.89·10-s + (−0.449 − 0.449i)11-s + 0.999·12-s + (2 − 3i)13-s + (−1.77 + 4.22i)14-s + (−2 + 2i)15-s + 5·16-s − 2·17-s + ⋯
L(s)  = 1  + (0.866 − 0.866i)2-s + 0.577i·3-s − 0.499i·4-s + (0.894 + 0.894i)5-s + (0.499 + 0.499i)6-s + (−0.925 + 0.377i)7-s + (0.433 + 0.433i)8-s − 0.333·9-s + 1.54·10-s + (−0.135 − 0.135i)11-s + 0.288·12-s + (0.554 − 0.832i)13-s + (−0.474 + 1.12i)14-s + (−0.516 + 0.516i)15-s + 1.25·16-s − 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.995 - 0.0934i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.995 - 0.0934i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06545 + 0.0967289i\)
\(L(\frac12)\) \(\approx\) \(2.06545 + 0.0967289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (2.44 - i)T \)
13 \( 1 + (-2 + 3i)T \)
good2 \( 1 + (-1.22 + 1.22i)T - 2iT^{2} \)
5 \( 1 + (-2 - 2i)T + 5iT^{2} \)
11 \( 1 + (0.449 + 0.449i)T + 11iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + (0.550 + 0.550i)T + 19iT^{2} \)
23 \( 1 + 8.89iT - 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 + (5.44 + 5.44i)T + 31iT^{2} \)
37 \( 1 + (-7.89 - 7.89i)T + 37iT^{2} \)
41 \( 1 + (4 + 4i)T + 41iT^{2} \)
43 \( 1 - 6.89iT - 43T^{2} \)
47 \( 1 + (-1.55 + 1.55i)T - 47iT^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (4.44 - 4.44i)T - 59iT^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + (6.34 - 6.34i)T - 67iT^{2} \)
71 \( 1 + (-2.44 + 2.44i)T - 71iT^{2} \)
73 \( 1 + (-7.89 + 7.89i)T - 73iT^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 + (-0.449 - 0.449i)T + 83iT^{2} \)
89 \( 1 + (-2 + 2i)T - 89iT^{2} \)
97 \( 1 + (-7.89 - 7.89i)T + 97iT^{2} \)
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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

−11.94769025847159881503660452233, −10.78144733336714730861205292662, −10.45076650335270780132697471901, −9.445454039739253751526251646545, −8.206372225224573619202379033956, −6.50603376781577016233424553217, −5.76977898245887487808557883787, −4.44096407113185634962311689770, −3.12835913266333210364713736769, −2.50355439749296610744771599728, 1.56181998106492146980760416093, 3.68541019113202767043097232560, 4.99964617995686551287702773918, 5.94083086041956919488307324277, 6.65106633912813387570766308701, 7.62031941751579119384226152799, 9.059579612730885656741436812444, 9.735058207660397834979223470240, 11.05171888360040324954728745124, 12.51022937305187909176898541643

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