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Properties

Label 2-272-16.13-c1-0-6
Degree $2$
Conductor $272$
Sign $-0.981 - 0.189i$
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  + (−0.205 + 1.39i)2-s + (1.44 + 1.44i)3-s + (−1.91 − 0.574i)4-s + (−1.55 + 1.55i)5-s + (−2.31 + 1.72i)6-s + 2.87i·7-s + (1.19 − 2.56i)8-s + 1.17i·9-s + (−1.85 − 2.49i)10-s + (0.709 − 0.709i)11-s + (−1.93 − 3.59i)12-s + (−1.15 − 1.15i)13-s + (−4.02 − 0.590i)14-s − 4.49·15-s + (3.33 + 2.20i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.145 + 0.989i)2-s + (0.834 + 0.834i)3-s + (−0.957 − 0.287i)4-s + (−0.694 + 0.694i)5-s + (−0.946 + 0.704i)6-s + 1.08i·7-s + (0.423 − 0.905i)8-s + 0.392i·9-s + (−0.586 − 0.788i)10-s + (0.213 − 0.213i)11-s + (−0.559 − 1.03i)12-s + (−0.320 − 0.320i)13-s + (−1.07 − 0.157i)14-s − 1.15·15-s + (0.834 + 0.550i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.981 - 0.189i$
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: \((2,\ 272,\ (\ :1/2),\ -0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111385 + 1.16726i\)
\(L(\frac12)\) \(\approx\) \(0.111385 + 1.16726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.205 - 1.39i)T \)
17 \( 1 + T \)
good3 \( 1 + (-1.44 - 1.44i)T + 3iT^{2} \)
5 \( 1 + (1.55 - 1.55i)T - 5iT^{2} \)
7 \( 1 - 2.87iT - 7T^{2} \)
11 \( 1 + (-0.709 + 0.709i)T - 11iT^{2} \)
13 \( 1 + (1.15 + 1.15i)T + 13iT^{2} \)
19 \( 1 + (-0.443 - 0.443i)T + 19iT^{2} \)
23 \( 1 - 3.88iT - 23T^{2} \)
29 \( 1 + (-1.60 - 1.60i)T + 29iT^{2} \)
31 \( 1 - 3.14T + 31T^{2} \)
37 \( 1 + (-1.86 + 1.86i)T - 37iT^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + (-8.83 + 8.83i)T - 43iT^{2} \)
47 \( 1 + 8.60T + 47T^{2} \)
53 \( 1 + (3.15 - 3.15i)T - 53iT^{2} \)
59 \( 1 + (-7.60 + 7.60i)T - 59iT^{2} \)
61 \( 1 + (-7.69 - 7.69i)T + 61iT^{2} \)
67 \( 1 + (-2.54 - 2.54i)T + 67iT^{2} \)
71 \( 1 - 2.42iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + (12.1 + 12.1i)T + 83iT^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + 2.79T + 97T^{2} \)
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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

−12.41127128489345549242094509448, −11.30603260466159877270443779722, −10.07138583303634431764358421702, −9.286281417927296554766090472178, −8.513866446784124305036070667381, −7.66346253751685234559883981757, −6.48838371465730216729438936726, −5.29222841786532611397142437418, −4.01758557240042403262845103661, −2.97640871997520696575982379302, 0.935022316026765515711472153870, 2.43780613374890356806691810726, 3.90715692416065188093107104309, 4.74500633100781755173862474039, 6.90281192267614103969503984731, 7.899388018756021339378650306624, 8.475091127865437472120332177778, 9.532582656088283015834160792201, 10.58715834313986812500911113239, 11.60466452572310038205167742214

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