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Properties

Label 2-272-16.5-c1-0-8
Degree $2$
Conductor $272$
Sign $-0.454 - 0.890i$
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  + (0.808 + 1.16i)2-s + (−0.0958 + 0.0958i)3-s + (−0.691 + 1.87i)4-s + (−0.447 − 0.447i)5-s + (−0.188 − 0.0336i)6-s + 0.608i·7-s + (−2.73 + 0.715i)8-s + 2.98i·9-s + (0.157 − 0.881i)10-s + (3.40 + 3.40i)11-s + (−0.113 − 0.246i)12-s + (−2.42 + 2.42i)13-s + (−0.706 + 0.492i)14-s + 0.0858·15-s + (−3.04 − 2.59i)16-s + 17-s + ⋯
L(s)  = 1  + (0.571 + 0.820i)2-s + (−0.0553 + 0.0553i)3-s + (−0.345 + 0.938i)4-s + (−0.200 − 0.200i)5-s + (−0.0770 − 0.0137i)6-s + 0.230i·7-s + (−0.967 + 0.253i)8-s + 0.993i·9-s + (0.0497 − 0.278i)10-s + (1.02 + 1.02i)11-s + (−0.0328 − 0.0710i)12-s + (−0.672 + 0.672i)13-s + (−0.188 + 0.131i)14-s + 0.0221·15-s + (−0.760 − 0.648i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ -0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783949 + 1.28052i\)
\(L(\frac12)\) \(\approx\) \(0.783949 + 1.28052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.808 - 1.16i)T \)
17 \( 1 - T \)
good3 \( 1 + (0.0958 - 0.0958i)T - 3iT^{2} \)
5 \( 1 + (0.447 + 0.447i)T + 5iT^{2} \)
7 \( 1 - 0.608iT - 7T^{2} \)
11 \( 1 + (-3.40 - 3.40i)T + 11iT^{2} \)
13 \( 1 + (2.42 - 2.42i)T - 13iT^{2} \)
19 \( 1 + (-2.29 + 2.29i)T - 19iT^{2} \)
23 \( 1 + 8.44iT - 23T^{2} \)
29 \( 1 + (1.91 - 1.91i)T - 29iT^{2} \)
31 \( 1 - 8.60T + 31T^{2} \)
37 \( 1 + (-2.39 - 2.39i)T + 37iT^{2} \)
41 \( 1 - 4.54iT - 41T^{2} \)
43 \( 1 + (2.24 + 2.24i)T + 43iT^{2} \)
47 \( 1 - 5.03T + 47T^{2} \)
53 \( 1 + (8.40 + 8.40i)T + 53iT^{2} \)
59 \( 1 + (-3.63 - 3.63i)T + 59iT^{2} \)
61 \( 1 + (4.10 - 4.10i)T - 61iT^{2} \)
67 \( 1 + (6.40 - 6.40i)T - 67iT^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 - 2.82T + 79T^{2} \)
83 \( 1 + (-10.0 + 10.0i)T - 83iT^{2} \)
89 \( 1 + 5.42iT - 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−12.18296880119283833160048113129, −11.74769450651718334054972989268, −10.21949206356669102698192823827, −9.142622373114176736556340544449, −8.192465247494108423069938869718, −7.17505317618713310178754901255, −6.34470644263064015059738693546, −4.84545386875165428006335468323, −4.37506439231050140980192789489, −2.51024989462559078659853048116, 1.08821497092078462060731150439, 3.17085747542493143653080296613, 3.87966464227075844893970150382, 5.45830220688667196517290942322, 6.31908896784462432506044071535, 7.64010716683390970544023576304, 9.136499915292979614624831450234, 9.748095205932126889718779789387, 10.90783435943536525328760548861, 11.76108098970475579450346742147

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