Properties

Label 2-273-91.76-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.0154 - 0.999i$
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  + (0.687 − 2.56i)2-s + (−0.866 + 0.5i)3-s + (−4.37 − 2.52i)4-s + (−1.17 + 1.17i)5-s + (0.687 + 2.56i)6-s + (−1.53 + 2.15i)7-s + (−5.73 + 5.73i)8-s + (0.499 − 0.866i)9-s + (2.21 + 3.83i)10-s + (−4.50 − 1.20i)11-s + 5.05·12-s + (−3.59 + 0.264i)13-s + (4.46 + 5.42i)14-s + (0.431 − 1.60i)15-s + (5.71 + 9.90i)16-s + (3.15 − 5.46i)17-s + ⋯
L(s)  = 1  + (0.486 − 1.81i)2-s + (−0.499 + 0.288i)3-s + (−2.18 − 1.26i)4-s + (−0.526 + 0.526i)5-s + (0.280 + 1.04i)6-s + (−0.581 + 0.813i)7-s + (−2.02 + 2.02i)8-s + (0.166 − 0.288i)9-s + (0.699 + 1.21i)10-s + (−1.35 − 0.363i)11-s + 1.45·12-s + (−0.997 + 0.0733i)13-s + (1.19 + 1.44i)14-s + (0.111 − 0.415i)15-s + (1.42 + 2.47i)16-s + (0.765 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0154 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0154 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.0154 - 0.999i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.0154 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104856 + 0.106491i\)
\(L(\frac12)\) \(\approx\) \(0.104856 + 0.106491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (1.53 - 2.15i)T \)
13 \( 1 + (3.59 - 0.264i)T \)
good2 \( 1 + (-0.687 + 2.56i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.17 - 1.17i)T - 5iT^{2} \)
11 \( 1 + (4.50 + 1.20i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.15 + 5.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.09 + 4.08i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.526 + 0.912i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.61 - 5.61i)T - 31iT^{2} \)
37 \( 1 + (-0.572 - 0.153i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.24 - 0.333i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (9.27 + 5.35i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.85 - 2.85i)T + 47iT^{2} \)
53 \( 1 + 0.398T + 53T^{2} \)
59 \( 1 + (8.26 - 2.21i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.22 - 2.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.76 - 10.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.31 + 1.15i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.935 + 0.935i)T + 73iT^{2} \)
79 \( 1 + 0.927T + 79T^{2} \)
83 \( 1 + (-7.79 + 7.79i)T - 83iT^{2} \)
89 \( 1 + (-1.28 + 4.78i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.65 + 6.18i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.27351365702103683383615585280, −10.54508681660606524555499161080, −9.719416337077333612604716301553, −8.868402281321166656982213253851, −7.19801257586244181000856543776, −5.43671696755369752205380609689, −4.87706834620962207188930031345, −3.24035268547547333514435169942, −2.63184209096183678589335139321, −0.099913897273076668850245891746, 3.72170485394946124825530405751, 4.80661116017522710364231851305, 5.65462109530088606744377333102, 6.71734075295504101105964935988, 7.80535357432372049505271089460, 7.951888059906005450571143099076, 9.561472740220081644405909809862, 10.54599354793791736485735322075, 12.33430587226930277207335219812, 12.78071614443580578397821560441

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