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Properties

Label 2-273-91.59-c1-0-2
Degree $2$
Conductor $273$
Sign $0.999 + 0.0102i$
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  + (−0.745 − 0.745i)2-s + (−0.866 + 0.5i)3-s − 0.887i·4-s + (3.80 + 1.01i)5-s + (1.01 + 0.273i)6-s + (0.148 + 2.64i)7-s + (−2.15 + 2.15i)8-s + (0.499 − 0.866i)9-s + (−2.07 − 3.59i)10-s + (−0.913 − 0.244i)11-s + (0.443 + 0.768i)12-s + (−0.783 + 3.51i)13-s + (1.85 − 2.08i)14-s + (−3.80 + 1.01i)15-s + 1.43·16-s + 7.96·17-s + ⋯
L(s)  = 1  + (−0.527 − 0.527i)2-s + (−0.499 + 0.288i)3-s − 0.443i·4-s + (1.70 + 0.455i)5-s + (0.415 + 0.111i)6-s + (0.0562 + 0.998i)7-s + (−0.761 + 0.761i)8-s + (0.166 − 0.288i)9-s + (−0.656 − 1.13i)10-s + (−0.275 − 0.0738i)11-s + (0.128 + 0.221i)12-s + (−0.217 + 0.976i)13-s + (0.496 − 0.556i)14-s + (−0.982 + 0.263i)15-s + 0.359·16-s + 1.93·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0102i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.999 + 0.0102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06771 - 0.00549841i\)
\(L(\frac12)\) \(\approx\) \(1.06771 - 0.00549841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.148 - 2.64i)T \)
13 \( 1 + (0.783 - 3.51i)T \)
good2 \( 1 + (0.745 + 0.745i)T + 2iT^{2} \)
5 \( 1 + (-3.80 - 1.01i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.913 + 0.244i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 7.96T + 17T^{2} \)
19 \( 1 + (-0.451 - 1.68i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.93iT - 23T^{2} \)
29 \( 1 + (-1.71 + 2.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.17 + 4.37i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.88 - 6.88i)T - 37iT^{2} \)
41 \( 1 + (0.117 + 0.437i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0936 + 0.0540i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.19 + 4.45i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.747 + 1.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.42 + 3.42i)T + 59iT^{2} \)
61 \( 1 + (5.73 + 3.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.17 + 4.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.68 - 10.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.95 + 1.05i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.473 - 0.820i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.26 - 8.26i)T - 83iT^{2} \)
89 \( 1 + (-3.79 - 3.79i)T + 89iT^{2} \)
97 \( 1 + (11.5 + 3.09i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.76271212340952665381321328026, −10.67685812299368460497085782656, −9.922421846863860257125656622278, −9.549402370534587936533109545197, −8.464653595896949905970324202610, −6.55529936732228860623455928445, −5.80579788635801424004895824968, −5.13538899294242968944417384730, −2.76391807245188767088472816826, −1.68778429458355071862806716108, 1.21534177864321627613818435261, 3.23147138932882098117204477103, 5.17537273317836168430069970562, 5.90377038498075117855202613587, 7.14499658425658182384680183024, 7.80758955057195727945344245543, 9.100295605226385614969010959165, 9.974471650655807850735020107085, 10.58192543515899546967752997259, 12.17717151558087862310105947751

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