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Properties

Label 2-273-91.34-c1-0-5
Degree $2$
Conductor $273$
Sign $-0.601 - 0.798i$
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 + (−1.62 + 2.08i)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 + (−0.613 + 0.789i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.601 - 0.798i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.601 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.885634 + 1.77673i\)
\(L(\frac12)\) \(\approx\) \(0.885634 + 1.77673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.08i)T \)
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

−12.23786624002363485699847114839, −11.87968991193846831839530377226, −10.69961547746930957050342724490, −8.740818123449213155970227410459, −8.077035361749134201759649929006, −6.85258641305949729358356320737, −6.44985664907841095811509925868, −5.48128627018883790254681317547, −3.69686936531713600083718489598, −3.21692809015273868031947438661, 1.22855893337502783242407668102, 3.45235112331291947061426649707, 4.03956734621898412742612052888, 4.79237664650401256969333699709, 6.10561831470801046225225520014, 7.66281234545168895787676482632, 9.158366637837715076169716669609, 9.904464417763890927885976047402, 10.97935409318735311146152784775, 11.68438498670414881445007518130

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