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Properties

Label 2-273-273.233-c2-0-69
Degree $2$
Conductor $273$
Sign $-0.516 - 0.856i$
Analytic cond. $7.43871$
Root an. cond. $2.72740$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 − 2.96i)2-s + (0.447 − 2.96i)3-s + (−3.84 − 6.66i)4-s + (−3.81 + 6.60i)5-s + (−8.01 − 6.39i)6-s + (−4.11 − 5.66i)7-s − 12.6·8-s + (−8.59 − 2.65i)9-s + (13.0 + 22.5i)10-s + (1.31 + 2.27i)11-s + (−21.4 + 8.42i)12-s − 13·13-s + (−23.8 + 2.51i)14-s + (17.8 + 14.2i)15-s + (−6.19 + 10.7i)16-s + (23.8 − 13.7i)17-s + ⋯
L(s)  = 1  + (0.854 − 1.48i)2-s + (0.149 − 0.988i)3-s + (−0.961 − 1.66i)4-s + (−0.762 + 1.32i)5-s + (−1.33 − 1.06i)6-s + (−0.588 − 0.808i)7-s − 1.57·8-s + (−0.955 − 0.295i)9-s + (1.30 + 2.25i)10-s + (0.119 + 0.207i)11-s + (−1.78 + 0.702i)12-s − 13-s + (−1.70 + 0.179i)14-s + (1.19 + 0.951i)15-s + (−0.386 + 0.670i)16-s + (1.40 − 0.808i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.516 - 0.856i$
Analytic conductor: \(7.43871\)
Root analytic conductor: \(2.72740\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1),\ -0.516 - 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.681297 + 1.20631i\)
\(L(\frac12)\) \(\approx\) \(0.681297 + 1.20631i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.447 + 2.96i)T \)
7 \( 1 + (4.11 + 5.66i)T \)
13 \( 1 + 13T \)
good2 \( 1 + (-1.70 + 2.96i)T + (-2 - 3.46i)T^{2} \)
5 \( 1 + (3.81 - 6.60i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.31 - 2.27i)T + (-60.5 + 104. i)T^{2} \)
17 \( 1 + (-23.8 + 13.7i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.27 - 0.736i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (28.1 + 16.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 38.7iT - 841T^{2} \)
31 \( 1 + (-13.6 + 7.86i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-5.09 - 2.94i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 47.3T + 1.68e3T^{2} \)
43 \( 1 + 12.6T + 1.84e3T^{2} \)
47 \( 1 + (11.4 - 19.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-78.1 + 45.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-18.1 - 31.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-32.6 + 56.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (8.13 - 4.69i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 94.6T + 5.04e3T^{2} \)
73 \( 1 + (92.0 - 53.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (18.4 - 31.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 11.2T + 6.88e3T^{2} \)
89 \( 1 + (-77.0 + 133. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 22.6iT - 9.40e3T^{2} \)
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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

−11.50012285516248842241053300230, −10.27661323577387185929929171081, −9.807188040600929407255708582849, −7.79101624182636538865801465616, −7.13948778478697925926603346008, −5.94402333358319460725304003254, −4.23786827034173072189943510218, −3.20267181747593274627891463265, −2.42345411902072511851381860115, −0.51228019158960999593241621852, 3.39665008954503015173765060129, 4.33501821593976963082825035667, 5.30767557819184959309001702439, 5.86682990109857787930014009583, 7.52149893490036711549567660651, 8.360575272423698784863793060227, 9.039854427753360796791360975340, 10.10741681081075173061199709685, 11.97219290084257196001384205986, 12.32190636472163134866534395072

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