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Properties

Label 2-273-273.254-c1-0-30
Degree $2$
Conductor $273$
Sign $-0.521 + 0.852i$
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  + (1.12 + 0.301i)2-s + (−0.956 − 1.44i)3-s + (−0.555 − 0.320i)4-s + (0.0307 + 0.114i)5-s + (−0.641 − 1.91i)6-s + (−1.83 − 1.90i)7-s + (−2.17 − 2.17i)8-s + (−1.16 + 2.76i)9-s + 0.138i·10-s + (−2.30 − 2.30i)11-s + (0.0684 + 1.10i)12-s + (2.91 − 2.12i)13-s + (−1.48 − 2.70i)14-s + (0.136 − 0.154i)15-s + (−1.15 − 1.99i)16-s + (−0.665 + 1.15i)17-s + ⋯
L(s)  = 1  + (0.795 + 0.213i)2-s + (−0.552 − 0.833i)3-s + (−0.277 − 0.160i)4-s + (0.0137 + 0.0512i)5-s + (−0.261 − 0.781i)6-s + (−0.692 − 0.721i)7-s + (−0.769 − 0.769i)8-s + (−0.389 + 0.920i)9-s + 0.0437i·10-s + (−0.695 − 0.695i)11-s + (0.0197 + 0.320i)12-s + (0.808 − 0.589i)13-s + (−0.397 − 0.722i)14-s + (0.0351 − 0.0397i)15-s + (−0.288 − 0.498i)16-s + (−0.161 + 0.279i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.521 + 0.852i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.521 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.499529 - 0.891299i\)
\(L(\frac12)\) \(\approx\) \(0.499529 - 0.891299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.956 + 1.44i)T \)
7 \( 1 + (1.83 + 1.90i)T \)
13 \( 1 + (-2.91 + 2.12i)T \)
good2 \( 1 + (-1.12 - 0.301i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.0307 - 0.114i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.30 + 2.30i)T + 11iT^{2} \)
17 \( 1 + (0.665 - 1.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.29 - 3.29i)T + 19iT^{2} \)
23 \( 1 + (1.93 + 3.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.77 + 2.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.08 + 7.79i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.91 - 0.512i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.06 - 11.4i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.721 - 0.416i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.90 + 1.85i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.67 + 3.27i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.47 - 1.19i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 8.26T + 61T^{2} \)
67 \( 1 + (0.646 + 0.646i)T + 67iT^{2} \)
71 \( 1 + (15.9 + 4.26i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (12.6 + 3.38i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.52 + 2.64i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.83 + 2.83i)T - 83iT^{2} \)
89 \( 1 + (-1.62 + 6.05i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.43 + 12.8i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.81312849317789678029758620343, −10.72316014980145373955963090382, −9.929376812813607251756421317222, −8.458197459311975260406848335720, −7.46135338484490083168554334603, −6.13802038992170108387972847682, −5.84670404611160907556813927599, −4.38880627052353367008810215164, −3.07884050398532385230205953115, −0.66074949505871222664618638814, 2.86551407212662127927154423739, 3.92808870472883223357255310572, 5.09386744791605447855287923285, 5.71575726019330579337785043046, 7.06144147500434702326435395054, 8.877911669067198384013740394127, 9.234944397284796919060963681759, 10.47379781141772191193890044045, 11.49975555252397275278633546493, 12.21832893379938632508456160055

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