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Properties

Label 2-273-91.19-c1-0-4
Degree $2$
Conductor $273$
Sign $0.603 + 0.797i$
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  + (−2.64 − 0.707i)2-s + i·3-s + (4.74 + 2.73i)4-s + (−0.792 + 0.212i)5-s + (0.707 − 2.64i)6-s + (−1.19 − 2.36i)7-s + (−6.71 − 6.71i)8-s − 9-s + 2.24·10-s + (0.566 + 0.566i)11-s + (−2.73 + 4.74i)12-s + (2.33 − 2.75i)13-s + (1.48 + 7.07i)14-s + (−0.212 − 0.792i)15-s + (7.51 + 13.0i)16-s + (0.145 − 0.251i)17-s + ⋯
L(s)  = 1  + (−1.86 − 0.500i)2-s + 0.577i·3-s + (2.37 + 1.36i)4-s + (−0.354 + 0.0949i)5-s + (0.288 − 1.07i)6-s + (−0.451 − 0.892i)7-s + (−2.37 − 2.37i)8-s − 0.333·9-s + 0.708·10-s + (0.170 + 0.170i)11-s + (−0.790 + 1.36i)12-s + (0.646 − 0.763i)13-s + (0.397 + 1.89i)14-s + (−0.0547 − 0.204i)15-s + (1.87 + 3.25i)16-s + (0.0352 − 0.0610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.603 + 0.797i$
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: \((2,\ 273,\ (\ :1/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427587 - 0.212666i\)
\(L(\frac12)\) \(\approx\) \(0.427587 - 0.212666i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 2.36i)T \)
13 \( 1 + (-2.33 + 2.75i)T \)
good2 \( 1 + (2.64 + 0.707i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.792 - 0.212i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.566 - 0.566i)T + 11iT^{2} \)
17 \( 1 + (-0.145 + 0.251i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.50 - 1.50i)T + 19iT^{2} \)
23 \( 1 + (-8.07 + 4.66i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.87 + 4.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.35 + 5.03i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.591 + 2.20i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-8.80 + 2.35i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.71 + 1.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.25 - 4.70i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.69 + 4.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.22 - 12.0i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.84iT - 61T^{2} \)
67 \( 1 + (-8.97 + 8.97i)T - 67iT^{2} \)
71 \( 1 + (6.43 + 1.72i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (7.74 + 2.07i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.75 + 4.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.51 + 3.51i)T + 83iT^{2} \)
89 \( 1 + (4.25 + 1.14i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.20 - 11.9i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.16577130641869739552819668087, −10.70819125835158650571166571938, −9.849088240083111095823100526812, −9.128845703898951513047847611298, −8.052301036488495317902388803724, −7.32920624493043920278239981728, −6.18221471012584674284947397325, −3.95250298693294851909012407919, −2.84072496651146029154150945701, −0.76633915566654586537372124482, 1.29580226617473387344306893111, 2.85171840956009931272116481416, 5.54395770287178711495761847181, 6.55647023866949542498526936420, 7.25415183291981489290292957760, 8.417922539484760404126063669165, 8.952349894235905699697105914419, 9.762654868505987795474015311047, 11.10524443749440758935779200467, 11.59043792113054934018143427624

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