Properties

Label 2-272-16.13-c1-0-5
Degree $2$
Conductor $272$
Sign $-0.858 - 0.512i$
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  + (−1.20 + 0.742i)2-s + (1.79 + 1.79i)3-s + (0.897 − 1.78i)4-s + (−2.01 + 2.01i)5-s + (−3.50 − 0.829i)6-s − 1.70i·7-s + (0.246 + 2.81i)8-s + 3.47i·9-s + (0.927 − 3.91i)10-s + (−4.60 + 4.60i)11-s + (4.83 − 1.60i)12-s + (0.725 + 0.725i)13-s + (1.26 + 2.05i)14-s − 7.24·15-s + (−2.38 − 3.20i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.851 + 0.525i)2-s + (1.03 + 1.03i)3-s + (0.448 − 0.893i)4-s + (−0.899 + 0.899i)5-s + (−1.42 − 0.338i)6-s − 0.644i·7-s + (0.0872 + 0.996i)8-s + 1.15i·9-s + (0.293 − 1.23i)10-s + (−1.38 + 1.38i)11-s + (1.39 − 0.462i)12-s + (0.201 + 0.201i)13-s + (0.338 + 0.548i)14-s − 1.86·15-s + (−0.597 − 0.802i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.858 - 0.512i$
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: \((2,\ 272,\ (\ :1/2),\ -0.858 - 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239030 + 0.867104i\)
\(L(\frac12)\) \(\approx\) \(0.239030 + 0.867104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.742i)T \)
17 \( 1 - T \)
good3 \( 1 + (-1.79 - 1.79i)T + 3iT^{2} \)
5 \( 1 + (2.01 - 2.01i)T - 5iT^{2} \)
7 \( 1 + 1.70iT - 7T^{2} \)
11 \( 1 + (4.60 - 4.60i)T - 11iT^{2} \)
13 \( 1 + (-0.725 - 0.725i)T + 13iT^{2} \)
19 \( 1 + (-4.00 - 4.00i)T + 19iT^{2} \)
23 \( 1 + 3.44iT - 23T^{2} \)
29 \( 1 + (2.60 + 2.60i)T + 29iT^{2} \)
31 \( 1 - 5.07T + 31T^{2} \)
37 \( 1 + (4.56 - 4.56i)T - 37iT^{2} \)
41 \( 1 + 1.38iT - 41T^{2} \)
43 \( 1 + (5.11 - 5.11i)T - 43iT^{2} \)
47 \( 1 - 9.90T + 47T^{2} \)
53 \( 1 + (-2.21 + 2.21i)T - 53iT^{2} \)
59 \( 1 + (-9.24 + 9.24i)T - 59iT^{2} \)
61 \( 1 + (3.49 + 3.49i)T + 61iT^{2} \)
67 \( 1 + (-8.13 - 8.13i)T + 67iT^{2} \)
71 \( 1 + 0.0715iT - 71T^{2} \)
73 \( 1 + 1.83iT - 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + (-0.861 - 0.861i)T + 83iT^{2} \)
89 \( 1 - 6.80iT - 89T^{2} \)
97 \( 1 + 4.05T + 97T^{2} \)
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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

−12.01811395070995780464888168795, −10.78097250646375589507915993637, −10.18290596383126436933869150074, −9.646314861014311720063438244817, −8.235412809236193091280840239765, −7.72252973414189876983065868162, −6.86210880584780464635292705433, −5.04291198580652924002669115095, −3.83664046163248311225558231361, −2.55017995856421622994901347075, 0.816185675969577792486047795100, 2.54631617753764373386074506544, 3.47148552406776870105558458734, 5.44588570631376788446739376938, 7.22683206042093274059290401717, 7.948317392489289555205818351357, 8.567051107890735161728969420658, 9.107212051851640800467326975107, 10.63380192636447264513212129501, 11.69379932492672893884093632456

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