Properties

Label 2-273-273.44-c1-0-13
Degree $2$
Conductor $273$
Sign $0.992 + 0.126i$
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  + (0.578 − 0.155i)2-s + (−1.70 − 0.303i)3-s + (−1.42 + 0.820i)4-s + (0.609 − 0.163i)5-s + (−1.03 + 0.0889i)6-s + (2.12 − 1.57i)7-s + (−1.54 + 1.54i)8-s + (2.81 + 1.03i)9-s + (0.327 − 0.188i)10-s + (4.87 + 1.30i)11-s + (2.67 − 0.968i)12-s + (2.92 − 2.10i)13-s + (0.985 − 1.24i)14-s + (−1.08 + 0.0936i)15-s + (0.987 − 1.71i)16-s + (1.40 + 2.42i)17-s + ⋯
L(s)  = 1  + (0.409 − 0.109i)2-s + (−0.984 − 0.175i)3-s + (−0.710 + 0.410i)4-s + (0.272 − 0.0730i)5-s + (−0.422 + 0.0363i)6-s + (0.803 − 0.595i)7-s + (−0.545 + 0.545i)8-s + (0.938 + 0.344i)9-s + (0.103 − 0.0597i)10-s + (1.47 + 0.394i)11-s + (0.771 − 0.279i)12-s + (0.810 − 0.585i)13-s + (0.263 − 0.331i)14-s + (−0.281 + 0.0241i)15-s + (0.246 − 0.427i)16-s + (0.340 + 0.589i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.992 + 0.126i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.992 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18774 - 0.0751826i\)
\(L(\frac12)\) \(\approx\) \(1.18774 - 0.0751826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.303i)T \)
7 \( 1 + (-2.12 + 1.57i)T \)
13 \( 1 + (-2.92 + 2.10i)T \)
good2 \( 1 + (-0.578 + 0.155i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-0.609 + 0.163i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.87 - 1.30i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.40 - 2.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.846 + 3.16i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.0947 + 0.164i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.29iT - 29T^{2} \)
31 \( 1 + (-4.65 - 1.24i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-5.36 + 1.43i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.27 + 1.27i)T + 41iT^{2} \)
43 \( 1 - 1.11iT - 43T^{2} \)
47 \( 1 + (3.19 + 11.9i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (11.7 - 6.77i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.79 - 1.28i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.34 - 5.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.14 - 2.18i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.98 + 5.98i)T + 71iT^{2} \)
73 \( 1 + (-0.480 + 1.79i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.82 - 8.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.08 + 6.08i)T + 83iT^{2} \)
89 \( 1 + (-1.77 - 6.63i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.17 + 2.17i)T - 97iT^{2} \)
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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

−11.88218243460990912754044063786, −11.19996821732773569477994864289, −10.16034912481152789811039601659, −9.024416003942957449737794479428, −7.966437030211350087215542424267, −6.78791261134188469432209557764, −5.66859111404444317519022002072, −4.63593988529192080498165727511, −3.77049558918833680925399483059, −1.32259053818341247267441441218, 1.33816150424657050395625528028, 3.92331997877411888487126926405, 4.74938396716343572614882300053, 6.01822702994444647899659980046, 6.30483724204767847024591108575, 8.144809513544076999733483914749, 9.301765336494237046587817573842, 9.920022360332896537881009299255, 11.33495850667805107819827711045, 11.72461079479207796074753701720

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