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Properties

Label 2-272-17.2-c1-0-3
Degree $2$
Conductor $272$
Sign $0.997 + 0.0758i$
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  + (2.41 − i)3-s + (0.707 + 1.70i)5-s + (−0.414 + i)7-s + (2.70 − 2.70i)9-s + (1 + 0.414i)11-s + 1.41i·13-s + (3.41 + 3.41i)15-s + (−2.82 − 3i)17-s + (−3.41 − 3.41i)19-s + 2.82i·21-s + (−3.82 − 1.58i)23-s + (1.12 − 1.12i)25-s + (0.828 − 2i)27-s + (−1.70 − 4.12i)29-s + (3 − 1.24i)31-s + ⋯
L(s)  = 1  + (1.39 − 0.577i)3-s + (0.316 + 0.763i)5-s + (−0.156 + 0.377i)7-s + (0.902 − 0.902i)9-s + (0.301 + 0.124i)11-s + 0.392i·13-s + (0.881 + 0.881i)15-s + (−0.685 − 0.727i)17-s + (−0.783 − 0.783i)19-s + 0.617i·21-s + (−0.798 − 0.330i)23-s + (0.224 − 0.224i)25-s + (0.159 − 0.384i)27-s + (−0.317 − 0.765i)29-s + (0.538 − 0.223i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.997 + 0.0758i$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ 0.997 + 0.0758i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93772 - 0.0736294i\)
\(L(\frac12)\) \(\approx\) \(1.93772 - 0.0736294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (2.82 + 3i)T \)
good3 \( 1 + (-2.41 + i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.707 - 1.70i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.414 - i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1 - 0.414i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
19 \( 1 + (3.41 + 3.41i)T + 19iT^{2} \)
23 \( 1 + (3.82 + 1.58i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.70 + 4.12i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3 + 1.24i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.53 - 1.46i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.12 - 7.53i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-3.41 + 3.41i)T - 43iT^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + (-3.53 + 8.53i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 + (12.0 - 5i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.05 + 4.94i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.82 - 1.58i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.242 - 0.242i)T + 83iT^{2} \)
89 \( 1 - 9.41iT - 89T^{2} \)
97 \( 1 + (-2.46 - 5.94i)T + (-68.5 + 68.5i)T^{2} \)
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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

−12.02072775979417077269982405828, −10.93388494272437834974763789642, −9.720438441590880696923915964725, −8.988206897425960613572984067446, −8.092471997477409696003280294666, −7.00257325808556683222491431261, −6.28561178043512675991772289175, −4.38150992762983267362319106527, −2.90590682010912757027268591678, −2.14864973990715317652361619263, 1.91407434735326771740838176881, 3.49566776552730642040179248695, 4.34518154935611927329300234271, 5.76537630384345944888022446326, 7.24507130110391551629166244178, 8.573018150890803681354530335662, 8.744649954215858435559625316422, 9.951360375661317932386623118492, 10.58011584697175319456427712351, 12.11857676414801084560969025568

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