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Properties

Label 2-272-17.4-c3-0-17
Degree $2$
Conductor $272$
Sign $0.999 + 0.0318i$
Analytic cond. $16.0485$
Root an. cond. $4.00606$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.657 − 0.657i)3-s + (13.8 − 13.8i)5-s + (14.0 + 14.0i)7-s + 26.1i·9-s + (29.5 + 29.5i)11-s − 19.0·13-s − 18.1i·15-s + (44.8 − 53.8i)17-s + 149. i·19-s + 18.4·21-s + (−80.3 − 80.3i)23-s − 257. i·25-s + (34.9 + 34.9i)27-s + (104. − 104. i)29-s + (73.6 − 73.6i)31-s + ⋯
L(s)  = 1  + (0.126 − 0.126i)3-s + (1.23 − 1.23i)5-s + (0.756 + 0.756i)7-s + 0.968i·9-s + (0.808 + 0.808i)11-s − 0.406·13-s − 0.312i·15-s + (0.640 − 0.768i)17-s + 1.80i·19-s + 0.191·21-s + (−0.728 − 0.728i)23-s − 2.06i·25-s + (0.248 + 0.248i)27-s + (0.666 − 0.666i)29-s + (0.426 − 0.426i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.999 + 0.0318i$
Analytic conductor: \(16.0485\)
Root analytic conductor: \(4.00606\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :3/2),\ 0.999 + 0.0318i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.647645575\)
\(L(\frac12)\) \(\approx\) \(2.647645575\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-44.8 + 53.8i)T \)
good3 \( 1 + (-0.657 + 0.657i)T - 27iT^{2} \)
5 \( 1 + (-13.8 + 13.8i)T - 125iT^{2} \)
7 \( 1 + (-14.0 - 14.0i)T + 343iT^{2} \)
11 \( 1 + (-29.5 - 29.5i)T + 1.33e3iT^{2} \)
13 \( 1 + 19.0T + 2.19e3T^{2} \)
19 \( 1 - 149. iT - 6.85e3T^{2} \)
23 \( 1 + (80.3 + 80.3i)T + 1.21e4iT^{2} \)
29 \( 1 + (-104. + 104. i)T - 2.43e4iT^{2} \)
31 \( 1 + (-73.6 + 73.6i)T - 2.97e4iT^{2} \)
37 \( 1 + (196. - 196. i)T - 5.06e4iT^{2} \)
41 \( 1 + (-283. - 283. i)T + 6.89e4iT^{2} \)
43 \( 1 + 20.5iT - 7.95e4T^{2} \)
47 \( 1 + 81.5T + 1.03e5T^{2} \)
53 \( 1 + 626. iT - 1.48e5T^{2} \)
59 \( 1 + 301. iT - 2.05e5T^{2} \)
61 \( 1 + (-352. - 352. i)T + 2.26e5iT^{2} \)
67 \( 1 + 924.T + 3.00e5T^{2} \)
71 \( 1 + (-748. + 748. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-4.67 + 4.67i)T - 3.89e5iT^{2} \)
79 \( 1 + (364. + 364. i)T + 4.93e5iT^{2} \)
83 \( 1 + 772. iT - 5.71e5T^{2} \)
89 \( 1 + 221.T + 7.04e5T^{2} \)
97 \( 1 + (7.27 - 7.27i)T - 9.12e5iT^{2} \)
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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

−11.85930711360463764216067533239, −10.14433648477064291932772459617, −9.665786563668854648130252129004, −8.521394242134600809584403431141, −7.87888190689127190631406560990, −6.22688199571967314616030342396, −5.25887172065036425742830833436, −4.54936648082913044662101550406, −2.23948073979511915409729156979, −1.47821202507680714715295706679, 1.23671652017196250041802484227, 2.80282391017537379112513308059, 3.95541285928726700975019740421, 5.58277177154807319796848440772, 6.54900346259609890315635030586, 7.29268206167097205292432117110, 8.801926635821145098300520220341, 9.646084239398719576991312903826, 10.59193171117278571515332174176, 11.18830622914006511473709318425

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