↑

Properties

Label 2-273-39.5-c1-0-24
Degree $2$
Conductor $273$
Sign $-0.896 + 0.442i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.150 + 0.150i)2-s + (−1.22 − 1.22i)3-s − 1.95i·4-s + (−0.482 − 0.482i)5-s + (−0.000996 − 0.369i)6-s + (−0.707 − 0.707i)7-s + (0.596 − 0.596i)8-s + (0.0161 + 2.99i)9-s − 0.145i·10-s + (−2.26 + 2.26i)11-s + (−2.38 + 2.40i)12-s + (−0.580 − 3.55i)13-s − 0.213i·14-s + (0.00318 + 1.18i)15-s − 3.72·16-s − 6.99·17-s + ⋯
L(s)  = 1  + (0.106 + 0.106i)2-s + (−0.709 − 0.705i)3-s − 0.977i·4-s + (−0.215 − 0.215i)5-s + (−0.000406 − 0.150i)6-s + (−0.267 − 0.267i)7-s + (0.210 − 0.210i)8-s + (0.00539 + 0.999i)9-s − 0.0460i·10-s + (−0.683 + 0.683i)11-s + (−0.689 + 0.692i)12-s + (−0.161 − 0.986i)13-s − 0.0570i·14-s + (0.000822 + 0.305i)15-s − 0.932·16-s − 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.896 + 0.442i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.896 + 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155064 - 0.664610i\)
\(L(\frac12)\) \(\approx\) \(0.155064 - 0.664610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 + 1.22i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.580 + 3.55i)T \)
good2 \( 1 + (-0.150 - 0.150i)T + 2iT^{2} \)
5 \( 1 + (0.482 + 0.482i)T + 5iT^{2} \)
11 \( 1 + (2.26 - 2.26i)T - 11iT^{2} \)
17 \( 1 + 6.99T + 17T^{2} \)
19 \( 1 + (-1.76 + 1.76i)T - 19iT^{2} \)
23 \( 1 - 5.46T + 23T^{2} \)
29 \( 1 - 2.64iT - 29T^{2} \)
31 \( 1 + (-2.48 + 2.48i)T - 31iT^{2} \)
37 \( 1 + (-0.194 - 0.194i)T + 37iT^{2} \)
41 \( 1 + (2.65 + 2.65i)T + 41iT^{2} \)
43 \( 1 - 3.03iT - 43T^{2} \)
47 \( 1 + (-7.69 + 7.69i)T - 47iT^{2} \)
53 \( 1 + 6.50iT - 53T^{2} \)
59 \( 1 + (-6.12 + 6.12i)T - 59iT^{2} \)
61 \( 1 + 9.34T + 61T^{2} \)
67 \( 1 + (-5.81 + 5.81i)T - 67iT^{2} \)
71 \( 1 + (-8.10 - 8.10i)T + 71iT^{2} \)
73 \( 1 + (-6.20 - 6.20i)T + 73iT^{2} \)
79 \( 1 - 3.96T + 79T^{2} \)
83 \( 1 + (9.36 + 9.36i)T + 83iT^{2} \)
89 \( 1 + (8.54 - 8.54i)T - 89iT^{2} \)
97 \( 1 + (-7.88 + 7.88i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.35876844463896525720765130119, −10.67640515080191317247836786886, −9.859571238292673409707855382776, −8.517586170956087611061716509211, −7.24380082135415334280518813182, −6.57110901991135304973873795119, −5.35522734277779582690334861545, −4.63978150195317697658397061487, −2.33145770839218674732178320128, −0.53234205290960414845968088686, 2.81034690034517343165456895348, 3.98111613852128092377687174063, 4.99300039973578652714315008388, 6.36397747330494663700336300927, 7.30523508104537281464173310693, 8.688493209020102524452262434699, 9.343204536662993155495162677306, 10.78265784696280964542048281908, 11.31366326812490573169962463726, 12.11288191631780599669264858274

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