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Properties

Label 2-273-13.12-c1-0-5
Degree $2$
Conductor $273$
Sign $-0.832 - 0.554i$
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  + 2i·2-s + 3-s − 2·4-s + 3i·5-s + 2i·6-s − i·7-s + 9-s − 6·10-s − 2·12-s + (2 − 3i)13-s + 2·14-s + 3i·15-s − 4·16-s − 2·17-s + 2i·18-s + i·19-s + ⋯
L(s)  = 1  + 1.41i·2-s + 0.577·3-s − 4-s + 1.34i·5-s + 0.816i·6-s − 0.377i·7-s + 0.333·9-s − 1.89·10-s − 0.577·12-s + (0.554 − 0.832i)13-s + 0.534·14-s + 0.774i·15-s − 16-s − 0.485·17-s + 0.471i·18-s + 0.229i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.434397 + 1.43471i\)
\(L(\frac12)\) \(\approx\) \(0.434397 + 1.43471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + iT \)
13 \( 1 + (-2 + 3i)T \)
good2 \( 1 - 2iT - 2T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 5iT - 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 15T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + 13iT - 97T^{2} \)
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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

−12.53711924879391049248068412288, −11.00019349193892584344335973713, −10.44237773289844884506708290251, −9.091325888207403476248922949636, −8.106523840292540468554590842384, −7.29523439370080668495912550963, −6.60369478835799314994114036829, −5.58698705029836725130809403607, −4.00210781752750486635416871863, −2.65376140931775816657558005927, 1.25303551678348609773086333816, 2.51396560601296405833713333615, 3.98036133172615340916797878280, 4.81595767110589909730428898467, 6.47642883402419071704485276541, 8.132154481969938936747522698136, 9.012452303660284311020256051581, 9.495803727925246431264251006851, 10.65135977649544478028872182113, 11.75942999023472771407128015184

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