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Properties

Label 2-273-7.4-c3-0-23
Degree $2$
Conductor $273$
Sign $0.999 + 0.0394i$
Analytic cond. $16.1075$
Root an. cond. $4.01341$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.331 + 0.574i)2-s + (−1.5 + 2.59i)3-s + (3.78 − 6.54i)4-s + (5.49 + 9.51i)5-s − 1.98·6-s + (15.8 − 9.64i)7-s + 10.3·8-s + (−4.5 − 7.79i)9-s + (−3.64 + 6.30i)10-s + (18.7 − 32.4i)11-s + (11.3 + 19.6i)12-s − 13·13-s + (10.7 + 5.88i)14-s − 32.9·15-s + (−26.8 − 46.4i)16-s + (3.25 − 5.64i)17-s + ⋯
L(s)  = 1  + (0.117 + 0.202i)2-s + (−0.288 + 0.499i)3-s + (0.472 − 0.818i)4-s + (0.491 + 0.851i)5-s − 0.135·6-s + (0.853 − 0.520i)7-s + 0.455·8-s + (−0.166 − 0.288i)9-s + (−0.115 + 0.199i)10-s + (0.513 − 0.889i)11-s + (0.272 + 0.472i)12-s − 0.277·13-s + (0.205 + 0.112i)14-s − 0.567·15-s + (−0.419 − 0.725i)16-s + (0.0464 − 0.0804i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0394i$
Analytic conductor: \(16.1075\)
Root analytic conductor: \(4.01341\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :3/2),\ 0.999 + 0.0394i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.427981484\)
\(L(\frac12)\) \(\approx\) \(2.427981484\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 + (-15.8 + 9.64i)T \)
13 \( 1 + 13T \)
good2 \( 1 + (-0.331 - 0.574i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-5.49 - 9.51i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-18.7 + 32.4i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-3.25 + 5.64i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (3.65 + 6.32i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (9.17 + 15.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 283.T + 2.43e4T^{2} \)
31 \( 1 + (94.2 - 163. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (47.5 + 82.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 262.T + 6.89e4T^{2} \)
43 \( 1 - 63.3T + 7.95e4T^{2} \)
47 \( 1 + (-143. - 249. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-52.4 + 90.8i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (134. - 233. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-328. - 568. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (22.4 - 38.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 382.T + 3.57e5T^{2} \)
73 \( 1 + (22.6 - 39.2i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-293. - 507. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 90.0T + 5.71e5T^{2} \)
89 \( 1 + (481. + 833. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.28e3T + 9.12e5T^{2} \)
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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

−11.07856500365732766413436099594, −10.66180773970090779699045925932, −9.893245325421212994520938173102, −8.639696801529144745069327801892, −7.22975447968804490366778423771, −6.38767719651078951197869963034, −5.48079902433891453377581924083, −4.34568484487294809658336054929, −2.70440791664186003051443262104, −1.09022206722294800858893695483, 1.43383820591970581646914326675, 2.42600028038690359419539745941, 4.28849353960563309860821166330, 5.27444162056172661495755676576, 6.55442353964235416375125235987, 7.62902809544365690216087418111, 8.457127818040972690548742167608, 9.450820887434739307680994845659, 10.78344746114235277506385043565, 11.82488403599734694846665954482

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