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Properties

Label 2-272-16.13-c1-0-18
Degree $2$
Conductor $272$
Sign $-0.921 + 0.387i$
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  + (−1.41 + 0.00186i)2-s + (−0.274 − 0.274i)3-s + (1.99 − 0.00528i)4-s + (−2.33 + 2.33i)5-s + (0.389 + 0.388i)6-s + 0.445i·7-s + (−2.82 + 0.0112i)8-s − 2.84i·9-s + (3.30 − 3.31i)10-s + (−0.541 + 0.541i)11-s + (−0.551 − 0.548i)12-s + (−3.22 − 3.22i)13-s + (−0.000833 − 0.630i)14-s + 1.28·15-s + (3.99 − 0.0211i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00132i)2-s + (−0.158 − 0.158i)3-s + (0.999 − 0.00264i)4-s + (−1.04 + 1.04i)5-s + (0.158 + 0.158i)6-s + 0.168i·7-s + (−0.999 + 0.00396i)8-s − 0.949i·9-s + (1.04 − 1.04i)10-s + (−0.163 + 0.163i)11-s + (−0.159 − 0.158i)12-s + (−0.894 − 0.894i)13-s + (−0.000222 − 0.168i)14-s + 0.332·15-s + (0.999 − 0.00528i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.921 + 0.387i$
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: \((2,\ 272,\ (\ :1/2),\ -0.921 + 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0153607 - 0.0761705i\)
\(L(\frac12)\) \(\approx\) \(0.0153607 - 0.0761705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.00186i)T \)
17 \( 1 + T \)
good3 \( 1 + (0.274 + 0.274i)T + 3iT^{2} \)
5 \( 1 + (2.33 - 2.33i)T - 5iT^{2} \)
7 \( 1 - 0.445iT - 7T^{2} \)
11 \( 1 + (0.541 - 0.541i)T - 11iT^{2} \)
13 \( 1 + (3.22 + 3.22i)T + 13iT^{2} \)
19 \( 1 + (5.38 + 5.38i)T + 19iT^{2} \)
23 \( 1 - 7.35iT - 23T^{2} \)
29 \( 1 + (4.91 + 4.91i)T + 29iT^{2} \)
31 \( 1 + 4.01T + 31T^{2} \)
37 \( 1 + (2.25 - 2.25i)T - 37iT^{2} \)
41 \( 1 + 8.61iT - 41T^{2} \)
43 \( 1 + (2.61 - 2.61i)T - 43iT^{2} \)
47 \( 1 - 0.997T + 47T^{2} \)
53 \( 1 + (-2.69 + 2.69i)T - 53iT^{2} \)
59 \( 1 + (8.74 - 8.74i)T - 59iT^{2} \)
61 \( 1 + (-7.55 - 7.55i)T + 61iT^{2} \)
67 \( 1 + (-9.29 - 9.29i)T + 67iT^{2} \)
71 \( 1 - 2.68iT - 71T^{2} \)
73 \( 1 + 0.113iT - 73T^{2} \)
79 \( 1 - 5.48T + 79T^{2} \)
83 \( 1 + (4.04 + 4.04i)T + 83iT^{2} \)
89 \( 1 + 12.6iT - 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−11.38958819193385792596498468933, −10.60931723160311279301585374933, −9.615674645682939325007375518855, −8.586194207767826482321417477616, −7.36686082414033697954587902901, −7.05399553495596153440946594795, −5.74613058997108759041601582552, −3.73778277275632912080502894884, −2.55240867109139727274775062741, −0.07742693805914477655838073514, 2.01849518020895850763191125338, 4.03870989355796850754557465009, 5.14288923709050637717257087586, 6.69162577652936359591783991086, 7.82227039607648332916808033106, 8.391049434796618839839993763398, 9.325687833593365296021937417832, 10.51702423278383171109051843625, 11.16302555986734476910229059812, 12.26066424095010372414034046325

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