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Properties

Label 2-272-17.4-c1-0-7
Degree $2$
Conductor $272$
Sign $0.155 + 0.987i$
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  + (2.30 − 2.30i)3-s + (−1 + i)5-s + (−2.30 − 2.30i)7-s − 7.60i·9-s + (0.302 + 0.302i)11-s + 2.60·13-s + 4.60i·15-s + (3.60 + 2i)17-s − 0.605i·19-s − 10.6·21-s + (4.30 + 4.30i)23-s + 3i·25-s + (−10.6 − 10.6i)27-s + (−1.60 + 1.60i)29-s + (−4.30 + 4.30i)31-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)3-s + (−0.447 + 0.447i)5-s + (−0.870 − 0.870i)7-s − 2.53i·9-s + (0.0912 + 0.0912i)11-s + 0.722·13-s + 1.18i·15-s + (0.874 + 0.485i)17-s − 0.138i·19-s − 2.31·21-s + (0.897 + 0.897i)23-s + 0.600i·25-s + (−2.04 − 2.04i)27-s + (−0.298 + 0.298i)29-s + (−0.772 + 0.772i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ 0.155 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27449 - 1.08919i\)
\(L(\frac12)\) \(\approx\) \(1.27449 - 1.08919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-3.60 - 2i)T \)
good3 \( 1 + (-2.30 + 2.30i)T - 3iT^{2} \)
5 \( 1 + (1 - i)T - 5iT^{2} \)
7 \( 1 + (2.30 + 2.30i)T + 7iT^{2} \)
11 \( 1 + (-0.302 - 0.302i)T + 11iT^{2} \)
13 \( 1 - 2.60T + 13T^{2} \)
19 \( 1 + 0.605iT - 19T^{2} \)
23 \( 1 + (-4.30 - 4.30i)T + 23iT^{2} \)
29 \( 1 + (1.60 - 1.60i)T - 29iT^{2} \)
31 \( 1 + (4.30 - 4.30i)T - 31iT^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 - 3.39iT - 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 + 5.21iT - 53T^{2} \)
59 \( 1 - 8.60iT - 59T^{2} \)
61 \( 1 + (6.21 + 6.21i)T + 61iT^{2} \)
67 \( 1 + 9.21T + 67T^{2} \)
71 \( 1 + (2.90 - 2.90i)T - 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + (-0.302 - 0.302i)T + 79iT^{2} \)
83 \( 1 + 17.8iT - 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 + (7.60 - 7.60i)T - 97iT^{2} \)
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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

−11.94382130345323478924035779170, −10.77077712968445983226376636423, −9.535876731746230767388555573197, −8.699005550782022807983658645531, −7.48034372988942799608834372717, −7.20957625579740575962523486905, −6.09456351405417423356479798435, −3.68384864010316048561795901796, −3.14730228605152724737707946001, −1.33524828462927850609317118263, 2.65589232211278194751509031991, 3.58389970938491681017938258885, 4.63640519130211044094167580414, 5.88719454224003115266301413947, 7.64132870599896640960844724827, 8.641763449724044649458301830270, 9.141518985647714591026377593088, 9.960136910874984526814785961484, 10.94282005207512694780307377611, 12.18676268846141860258765560672

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