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Properties

Label 2-272-16.13-c1-0-26
Degree $2$
Conductor $272$
Sign $0.573 + 0.819i$
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  + (1.27 − 0.611i)2-s + (0.148 + 0.148i)3-s + (1.25 − 1.55i)4-s + (−0.0348 + 0.0348i)5-s + (0.280 + 0.0985i)6-s − 1.59i·7-s + (0.642 − 2.75i)8-s − 2.95i·9-s + (−0.0231 + 0.0657i)10-s + (−2.14 + 2.14i)11-s + (0.417 − 0.0457i)12-s + (4.14 + 4.14i)13-s + (−0.976 − 2.03i)14-s − 0.0103·15-s + (−0.865 − 3.90i)16-s − 17-s + ⋯
L(s)  = 1  + (0.901 − 0.432i)2-s + (0.0857 + 0.0857i)3-s + (0.625 − 0.779i)4-s + (−0.0155 + 0.0155i)5-s + (0.114 + 0.0402i)6-s − 0.603i·7-s + (0.227 − 0.973i)8-s − 0.985i·9-s + (−0.00730 + 0.0207i)10-s + (−0.647 + 0.647i)11-s + (0.120 − 0.0132i)12-s + (1.15 + 1.15i)13-s + (−0.260 − 0.543i)14-s − 0.00267·15-s + (−0.216 − 0.976i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.573 + 0.819i$
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: \((2,\ 272,\ (\ :1/2),\ 0.573 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88898 - 0.983357i\)
\(L(\frac12)\) \(\approx\) \(1.88898 - 0.983357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 + 0.611i)T \)
17 \( 1 + T \)
good3 \( 1 + (-0.148 - 0.148i)T + 3iT^{2} \)
5 \( 1 + (0.0348 - 0.0348i)T - 5iT^{2} \)
7 \( 1 + 1.59iT - 7T^{2} \)
11 \( 1 + (2.14 - 2.14i)T - 11iT^{2} \)
13 \( 1 + (-4.14 - 4.14i)T + 13iT^{2} \)
19 \( 1 + (-3.11 - 3.11i)T + 19iT^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + (4.28 + 4.28i)T + 29iT^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 + (-5.91 + 5.91i)T - 37iT^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + (8.31 - 8.31i)T - 43iT^{2} \)
47 \( 1 + 3.71T + 47T^{2} \)
53 \( 1 + (-5.48 + 5.48i)T - 53iT^{2} \)
59 \( 1 + (-2.61 + 2.61i)T - 59iT^{2} \)
61 \( 1 + (-3.06 - 3.06i)T + 61iT^{2} \)
67 \( 1 + (-3.49 - 3.49i)T + 67iT^{2} \)
71 \( 1 + 3.25iT - 71T^{2} \)
73 \( 1 + 3.34iT - 73T^{2} \)
79 \( 1 - 3.31T + 79T^{2} \)
83 \( 1 + (4.33 + 4.33i)T + 83iT^{2} \)
89 \( 1 + 9.44iT - 89T^{2} \)
97 \( 1 - 5.37T + 97T^{2} \)
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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

−11.61338669944835888425804307734, −11.19623305094165241647008305129, −9.924323334502601176002879192995, −9.264141253243284445171745334510, −7.59331886760288498963805373864, −6.63145347643671914228226332353, −5.59492101051363152208119036907, −4.20688407683465884815424886356, −3.45042783367778510345517493427, −1.61956845784685530218536168802, 2.42192268666097228866083214299, 3.58377561383995441060705824731, 5.22199418784307811827453482989, 5.68551942571976595198831603337, 7.07881362813061475362437144841, 8.114553562931672365202912072096, 8.762527644236425748314879358756, 10.60563219367039902758946147935, 11.11305543875010277701473766865, 12.28350863894483697342924944526

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