Properties

Label 2-273-91.83-c1-0-16
Degree $2$
Conductor $273$
Sign $-0.385 + 0.922i$
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  + (1.75 − 1.75i)2-s i·3-s − 4.16i·4-s + (2.27 + 2.27i)5-s + (−1.75 − 1.75i)6-s + (−2.08 − 1.62i)7-s + (−3.80 − 3.80i)8-s − 9-s + 7.98·10-s + (3.22 + 3.22i)11-s − 4.16·12-s + (−3.60 − 0.106i)13-s + (−6.51 + 0.819i)14-s + (2.27 − 2.27i)15-s − 5.02·16-s − 4.17·17-s + ⋯
L(s)  = 1  + (1.24 − 1.24i)2-s − 0.577i·3-s − 2.08i·4-s + (1.01 + 1.01i)5-s + (−0.716 − 0.716i)6-s + (−0.789 − 0.613i)7-s + (−1.34 − 1.34i)8-s − 0.333·9-s + 2.52·10-s + (0.972 + 0.972i)11-s − 1.20·12-s + (−0.999 − 0.0294i)13-s + (−1.74 + 0.219i)14-s + (0.586 − 0.586i)15-s − 1.25·16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.385 + 0.922i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35915 - 2.04073i\)
\(L(\frac12)\) \(\approx\) \(1.35915 - 2.04073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.08 + 1.62i)T \)
13 \( 1 + (3.60 + 0.106i)T \)
good2 \( 1 + (-1.75 + 1.75i)T - 2iT^{2} \)
5 \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \)
11 \( 1 + (-3.22 - 3.22i)T + 11iT^{2} \)
17 \( 1 + 4.17T + 17T^{2} \)
19 \( 1 + (-0.774 - 0.774i)T + 19iT^{2} \)
23 \( 1 - 3.94iT - 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 + (-4.54 - 4.54i)T + 31iT^{2} \)
37 \( 1 + (7.34 + 7.34i)T + 37iT^{2} \)
41 \( 1 + (6.60 + 6.60i)T + 41iT^{2} \)
43 \( 1 - 5.39iT - 43T^{2} \)
47 \( 1 + (-7.77 + 7.77i)T - 47iT^{2} \)
53 \( 1 + 0.429T + 53T^{2} \)
59 \( 1 + (-1.29 + 1.29i)T - 59iT^{2} \)
61 \( 1 + 3.25iT - 61T^{2} \)
67 \( 1 + (-1.90 + 1.90i)T - 67iT^{2} \)
71 \( 1 + (3.74 - 3.74i)T - 71iT^{2} \)
73 \( 1 + (2.36 - 2.36i)T - 73iT^{2} \)
79 \( 1 - 3.64T + 79T^{2} \)
83 \( 1 + (-0.336 - 0.336i)T + 83iT^{2} \)
89 \( 1 + (11.2 - 11.2i)T - 89iT^{2} \)
97 \( 1 + (0.459 + 0.459i)T + 97iT^{2} \)
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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

−11.91882701811377606413702998544, −10.71856015756813175556211993375, −10.10168671636359595322312192933, −9.351402347669562285865940540739, −7.01536833412283547464091477540, −6.59903742718542508174748376571, −5.34031999833870204289233180041, −3.97801204225206324104780719780, −2.77461530427877415836363093137, −1.80775590129095974946344862483, 2.86072800290426622591154756869, 4.32998860938733321078788511649, 5.15999750891683806551586337607, 6.07709518316998100710416728620, 6.70093106615497918558366970008, 8.462041420648562639512362205865, 9.039127009265528631663870908840, 10.03761279499410079795195191219, 11.78891618717788528186385847308, 12.46832756979333209449892530355

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