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Properties

Label 2-272-16.13-c1-0-30
Degree $2$
Conductor $272$
Sign $-0.865 - 0.500i$
Analytic cond. $2.17193$
Root an. cond. $1.47374$
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  + (0.321 − 1.37i)2-s + (−2.20 − 2.20i)3-s + (−1.79 − 0.885i)4-s + (2.24 − 2.24i)5-s + (−3.74 + 2.33i)6-s − 3.15i·7-s + (−1.79 + 2.18i)8-s + 6.74i·9-s + (−2.36 − 3.80i)10-s + (2.18 − 2.18i)11-s + (2.00 + 5.91i)12-s + (1.93 + 1.93i)13-s + (−4.34 − 1.01i)14-s − 9.89·15-s + (2.43 + 3.17i)16-s − 17-s + ⋯
L(s)  = 1  + (0.227 − 0.973i)2-s + (−1.27 − 1.27i)3-s + (−0.896 − 0.442i)4-s + (1.00 − 1.00i)5-s + (−1.53 + 0.951i)6-s − 1.19i·7-s + (−0.634 + 0.772i)8-s + 2.24i·9-s + (−0.748 − 1.20i)10-s + (0.659 − 0.659i)11-s + (0.578 + 1.70i)12-s + (0.536 + 0.536i)13-s + (−1.16 − 0.270i)14-s − 2.55·15-s + (0.608 + 0.793i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.865 - 0.500i$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ -0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259582 + 0.967785i\)
\(L(\frac12)\) \(\approx\) \(0.259582 + 0.967785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.321 + 1.37i)T \)
17 \( 1 + T \)
good3 \( 1 + (2.20 + 2.20i)T + 3iT^{2} \)
5 \( 1 + (-2.24 + 2.24i)T - 5iT^{2} \)
7 \( 1 + 3.15iT - 7T^{2} \)
11 \( 1 + (-2.18 + 2.18i)T - 11iT^{2} \)
13 \( 1 + (-1.93 - 1.93i)T + 13iT^{2} \)
19 \( 1 + (-5.15 - 5.15i)T + 19iT^{2} \)
23 \( 1 - 0.195iT - 23T^{2} \)
29 \( 1 + (0.528 + 0.528i)T + 29iT^{2} \)
31 \( 1 + 6.11T + 31T^{2} \)
37 \( 1 + (-1.40 + 1.40i)T - 37iT^{2} \)
41 \( 1 + 4.43iT - 41T^{2} \)
43 \( 1 + (0.190 - 0.190i)T - 43iT^{2} \)
47 \( 1 - 3.42T + 47T^{2} \)
53 \( 1 + (-8.10 + 8.10i)T - 53iT^{2} \)
59 \( 1 + (8.44 - 8.44i)T - 59iT^{2} \)
61 \( 1 + (-0.714 - 0.714i)T + 61iT^{2} \)
67 \( 1 + (6.97 + 6.97i)T + 67iT^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 - 9.75iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + (-6.54 - 6.54i)T + 83iT^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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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

−11.50127978091883024946120104169, −10.73694513217635141913551001171, −9.734704704454331918774967624296, −8.601480743413030864785451491856, −7.23345533862900802013767982298, −5.97392123218435831219824644781, −5.41196635385728221687447931613, −4.03196180960064179914404176818, −1.65833379239036168267058544058, −0.955266944501218711267585836350, 3.17383907648099418112129262623, 4.66458404524654967041682103905, 5.63177123324909353926813177622, 6.11952905704649338545572882332, 7.09568395564861925004533403509, 9.073471386877672523887486611005, 9.483859761475350230754371490861, 10.46758861853251442718960487261, 11.45303441457067202391356919966, 12.32575819561799111851511844285

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