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Properties

Label 2-273-91.19-c1-0-17
Degree $2$
Conductor $273$
Sign $-0.733 + 0.680i$
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  + (0.446 + 0.119i)2-s + i·3-s + (−1.54 − 0.893i)4-s + (−2.09 + 0.562i)5-s + (−0.119 + 0.446i)6-s + (−1.26 − 2.32i)7-s + (−1.23 − 1.23i)8-s − 9-s − 1.00·10-s + (−4.08 − 4.08i)11-s + (0.893 − 1.54i)12-s + (2.90 + 2.13i)13-s + (−0.288 − 1.18i)14-s + (−0.562 − 2.09i)15-s + (1.38 + 2.39i)16-s + (0.0614 − 0.106i)17-s + ⋯
L(s)  = 1  + (0.315 + 0.0845i)2-s + 0.577i·3-s + (−0.773 − 0.446i)4-s + (−0.938 + 0.251i)5-s + (−0.0488 + 0.182i)6-s + (−0.479 − 0.877i)7-s + (−0.437 − 0.437i)8-s − 0.333·9-s − 0.317·10-s + (−1.23 − 1.23i)11-s + (0.257 − 0.446i)12-s + (0.805 + 0.592i)13-s + (−0.0772 − 0.317i)14-s + (−0.145 − 0.541i)15-s + (0.345 + 0.598i)16-s + (0.0149 − 0.0258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.733 + 0.680i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.733 + 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113034 - 0.287997i\)
\(L(\frac12)\) \(\approx\) \(0.113034 - 0.287997i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 2.32i)T \)
13 \( 1 + (-2.90 - 2.13i)T \)
good2 \( 1 + (-0.446 - 0.119i)T + (1.73 + i)T^{2} \)
5 \( 1 + (2.09 - 0.562i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.08 + 4.08i)T + 11iT^{2} \)
17 \( 1 + (-0.0614 + 0.106i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.29 + 2.29i)T + 19iT^{2} \)
23 \( 1 + (2.23 - 1.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.15 - 7.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.940 + 3.50i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.75 + 6.55i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.63 - 0.707i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.72 + 2.14i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.591 - 2.20i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.49 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.94 + 7.26i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 6.15iT - 61T^{2} \)
67 \( 1 + (7.99 - 7.99i)T - 67iT^{2} \)
71 \( 1 + (-7.44 - 1.99i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-12.0 - 3.23i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.34 + 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.76 - 4.76i)T + 83iT^{2} \)
89 \( 1 + (-0.966 - 0.259i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.38 + 8.88i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.10457312221951836652559603068, −10.85401815692914302397608598951, −9.683769349910507009053367364792, −8.699223590013146519323341531717, −7.74625636908807235732394718116, −6.37791513603704505530874960536, −5.27525755239253427167205638231, −4.03415163295725163725336880899, −3.42158394173312721320804868131, −0.21061998020521244736993388410, 2.57011530142023695237836725303, 3.88870137436524563536578292563, 5.05282894415339125153654548427, 6.16894473052232135925118666449, 7.82794308704507250274718870794, 8.096091907587724118284263342652, 9.240327142071189308757436209702, 10.39302219858265334585330674820, 11.81368743075759558341192607668, 12.37557437251568761852875050043

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