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Properties

Label 2-272-16.13-c1-0-25
Degree $2$
Conductor $272$
Sign $0.235 + 0.971i$
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  + (0.591 − 1.28i)2-s + (0.915 + 0.915i)3-s + (−1.30 − 1.51i)4-s + (2.93 − 2.93i)5-s + (1.71 − 0.634i)6-s + 1.91i·7-s + (−2.72 + 0.771i)8-s − 1.32i·9-s + (−2.03 − 5.51i)10-s + (−2.96 + 2.96i)11-s + (0.201 − 2.58i)12-s + (1.16 + 1.16i)13-s + (2.45 + 1.13i)14-s + 5.37·15-s + (−0.619 + 3.95i)16-s + 17-s + ⋯
L(s)  = 1  + (0.418 − 0.908i)2-s + (0.528 + 0.528i)3-s + (−0.650 − 0.759i)4-s + (1.31 − 1.31i)5-s + (0.701 − 0.259i)6-s + 0.723i·7-s + (−0.962 + 0.272i)8-s − 0.441i·9-s + (−0.643 − 1.74i)10-s + (−0.894 + 0.894i)11-s + (0.0580 − 0.745i)12-s + (0.321 + 0.321i)13-s + (0.657 + 0.302i)14-s + 1.38·15-s + (−0.154 + 0.987i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.235 + 0.971i$
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: \((2,\ 272,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52651 - 1.20131i\)
\(L(\frac12)\) \(\approx\) \(1.52651 - 1.20131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.591 + 1.28i)T \)
17 \( 1 - T \)
good3 \( 1 + (-0.915 - 0.915i)T + 3iT^{2} \)
5 \( 1 + (-2.93 + 2.93i)T - 5iT^{2} \)
7 \( 1 - 1.91iT - 7T^{2} \)
11 \( 1 + (2.96 - 2.96i)T - 11iT^{2} \)
13 \( 1 + (-1.16 - 1.16i)T + 13iT^{2} \)
19 \( 1 + (1.64 + 1.64i)T + 19iT^{2} \)
23 \( 1 - 0.425iT - 23T^{2} \)
29 \( 1 + (-4.17 - 4.17i)T + 29iT^{2} \)
31 \( 1 + 4.08T + 31T^{2} \)
37 \( 1 + (7.58 - 7.58i)T - 37iT^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 + (-4.70 + 4.70i)T - 43iT^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 + (1.05 - 1.05i)T - 53iT^{2} \)
59 \( 1 + (6.30 - 6.30i)T - 59iT^{2} \)
61 \( 1 + (6.38 + 6.38i)T + 61iT^{2} \)
67 \( 1 + (3.95 + 3.95i)T + 67iT^{2} \)
71 \( 1 - 7.48iT - 71T^{2} \)
73 \( 1 + 0.137iT - 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + (9.36 + 9.36i)T + 83iT^{2} \)
89 \( 1 + 3.78iT - 89T^{2} \)
97 \( 1 - 1.68T + 97T^{2} \)
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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

−12.07179637516697618454660965825, −10.50755738080881482107982012550, −9.765115394238596899831792856472, −9.098438697875161510389873565639, −8.513656655871312153503822869288, −6.26592179240806975059960045285, −5.21153173583690424277309430698, −4.52423594139995495860399367821, −2.85769243175883387662621961928, −1.64906091974799633536123633413, 2.41642604888447670163514799311, 3.50136724210261868124818845508, 5.38253798240969640712972996516, 6.18303671762093272873386909491, 7.19935457864013780510132696302, 7.87687663190405069411571331282, 9.012475511911690570924249019470, 10.37223938221981077019423101694, 10.82749002217119166598941826999, 12.63851991621306202024319815616

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