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Properties

Label 2-272-16.13-c1-0-22
Degree $2$
Conductor $272$
Sign $-0.957 + 0.287i$
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  + (−0.974 − 1.02i)2-s + (−1.15 − 1.15i)3-s + (−0.101 + 1.99i)4-s + (1.52 − 1.52i)5-s + (−0.0584 + 2.31i)6-s − 1.16i·7-s + (2.14 − 1.84i)8-s − 0.329i·9-s + (−3.05 − 0.0772i)10-s + (0.0215 − 0.0215i)11-s + (2.42 − 2.19i)12-s + (−0.154 − 0.154i)13-s + (−1.19 + 1.13i)14-s − 3.53·15-s + (−3.97 − 0.403i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.689 − 0.724i)2-s + (−0.667 − 0.667i)3-s + (−0.0505 + 0.998i)4-s + (0.683 − 0.683i)5-s + (−0.0238 + 0.943i)6-s − 0.439i·7-s + (0.758 − 0.651i)8-s − 0.109i·9-s + (−0.966 − 0.0244i)10-s + (0.00650 − 0.00650i)11-s + (0.700 − 0.632i)12-s + (−0.0428 − 0.0428i)13-s + (−0.318 + 0.302i)14-s − 0.912·15-s + (−0.994 − 0.100i)16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.957 + 0.287i$
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: \((2,\ 272,\ (\ :1/2),\ -0.957 + 0.287i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0979397 - 0.666971i\)
\(L(\frac12)\) \(\approx\) \(0.0979397 - 0.666971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.974 + 1.02i)T \)
17 \( 1 - T \)
good3 \( 1 + (1.15 + 1.15i)T + 3iT^{2} \)
5 \( 1 + (-1.52 + 1.52i)T - 5iT^{2} \)
7 \( 1 + 1.16iT - 7T^{2} \)
11 \( 1 + (-0.0215 + 0.0215i)T - 11iT^{2} \)
13 \( 1 + (0.154 + 0.154i)T + 13iT^{2} \)
19 \( 1 + (5.80 + 5.80i)T + 19iT^{2} \)
23 \( 1 - 2.25iT - 23T^{2} \)
29 \( 1 + (1.45 + 1.45i)T + 29iT^{2} \)
31 \( 1 + 7.04T + 31T^{2} \)
37 \( 1 + (-4.76 + 4.76i)T - 37iT^{2} \)
41 \( 1 - 2.52iT - 41T^{2} \)
43 \( 1 + (-4.98 + 4.98i)T - 43iT^{2} \)
47 \( 1 - 5.36T + 47T^{2} \)
53 \( 1 + (-5.25 + 5.25i)T - 53iT^{2} \)
59 \( 1 + (-4.81 + 4.81i)T - 59iT^{2} \)
61 \( 1 + (7.32 + 7.32i)T + 61iT^{2} \)
67 \( 1 + (-2.91 - 2.91i)T + 67iT^{2} \)
71 \( 1 + 1.34iT - 71T^{2} \)
73 \( 1 - 4.38iT - 73T^{2} \)
79 \( 1 - 0.873T + 79T^{2} \)
83 \( 1 + (-10.8 - 10.8i)T + 83iT^{2} \)
89 \( 1 - 5.90iT - 89T^{2} \)
97 \( 1 - 8.99T + 97T^{2} \)
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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

−11.36804022960541687139167503970, −10.72003943234291139931395298294, −9.486200818135305907764666881306, −8.904861550917976762461447097912, −7.57362054405007784387588064704, −6.68148907013583541702775663814, −5.41232226249889740876978106933, −3.95642846534026200727192960159, −2.09943329729675431159111780799, −0.70005940326503372384816136862, 2.17067015118340914930815862870, 4.37479396853057067573763898235, 5.70980302403241385968887410133, 6.12842772510232616829886229040, 7.42312710993281048985242112769, 8.566732100116953560963276328226, 9.608159982993905390811636365348, 10.49833219755828601772980115094, 10.79691594596234977024584203963, 12.12653557516897594769840939231

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