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Properties

Label 2-272-16.13-c1-0-14
Degree $2$
Conductor $272$
Sign $0.822 + 0.569i$
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  + (−1.37 − 0.349i)2-s + (0.724 + 0.724i)3-s + (1.75 + 0.957i)4-s + (0.100 − 0.100i)5-s + (−0.739 − 1.24i)6-s − 2.62i·7-s + (−2.07 − 1.92i)8-s − 1.95i·9-s + (−0.172 + 0.102i)10-s + (1.93 − 1.93i)11-s + (0.577 + 1.96i)12-s + (1.48 + 1.48i)13-s + (−0.918 + 3.60i)14-s + 0.144·15-s + (2.16 + 3.36i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.969 − 0.247i)2-s + (0.418 + 0.418i)3-s + (0.877 + 0.478i)4-s + (0.0447 − 0.0447i)5-s + (−0.301 − 0.508i)6-s − 0.993i·7-s + (−0.732 − 0.680i)8-s − 0.650i·9-s + (−0.0544 + 0.0323i)10-s + (0.583 − 0.583i)11-s + (0.166 + 0.567i)12-s + (0.412 + 0.412i)13-s + (−0.245 + 0.962i)14-s + 0.0374·15-s + (0.541 + 0.840i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.822 + 0.569i$
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: \((2,\ 272,\ (\ :1/2),\ 0.822 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.954495 - 0.298326i\)
\(L(\frac12)\) \(\approx\) \(0.954495 - 0.298326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.349i)T \)
17 \( 1 - T \)
good3 \( 1 + (-0.724 - 0.724i)T + 3iT^{2} \)
5 \( 1 + (-0.100 + 0.100i)T - 5iT^{2} \)
7 \( 1 + 2.62iT - 7T^{2} \)
11 \( 1 + (-1.93 + 1.93i)T - 11iT^{2} \)
13 \( 1 + (-1.48 - 1.48i)T + 13iT^{2} \)
19 \( 1 + (-2.90 - 2.90i)T + 19iT^{2} \)
23 \( 1 + 5.35iT - 23T^{2} \)
29 \( 1 + (0.891 + 0.891i)T + 29iT^{2} \)
31 \( 1 - 1.44T + 31T^{2} \)
37 \( 1 + (1.18 - 1.18i)T - 37iT^{2} \)
41 \( 1 - 4.71iT - 41T^{2} \)
43 \( 1 + (-7.44 + 7.44i)T - 43iT^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + (4.79 - 4.79i)T - 53iT^{2} \)
59 \( 1 + (6.48 - 6.48i)T - 59iT^{2} \)
61 \( 1 + (-1.30 - 1.30i)T + 61iT^{2} \)
67 \( 1 + (2.81 + 2.81i)T + 67iT^{2} \)
71 \( 1 + 6.09iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 2.30T + 79T^{2} \)
83 \( 1 + (-0.632 - 0.632i)T + 83iT^{2} \)
89 \( 1 - 11.0iT - 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−11.57028713721284108469541903663, −10.69745673761159961295920252917, −9.811354953372498046058568857645, −9.078690026700025353253812791481, −8.181792921824915812752415305078, −7.07853578479547034925939636118, −6.12643592139602849126561055478, −4.07808936186352672141170305767, −3.20449146535698044126534926821, −1.16305988228156779685952737931, 1.70892370879982073194435846885, 2.92565486492630876234865796231, 5.14044497874913494996824488538, 6.24515145728925914533294001539, 7.36795592181632605724756419522, 8.157810949596669755032156992227, 9.060377685967515994403248096419, 9.834914664318917935123319506513, 11.00546975789529451589168391278, 11.84327751466042134845027372480

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