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Properties

Label 2-272-16.5-c1-0-3
Degree $2$
Conductor $272$
Sign $-0.989 - 0.145i$
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  + (0.0871 + 1.41i)2-s + (0.0244 − 0.0244i)3-s + (−1.98 + 0.246i)4-s + (−1.10 − 1.10i)5-s + (0.0366 + 0.0323i)6-s + 5.11i·7-s + (−0.520 − 2.78i)8-s + 2.99i·9-s + (1.46 − 1.66i)10-s + (−3.11 − 3.11i)11-s + (−0.0425 + 0.0545i)12-s + (−1.02 + 1.02i)13-s + (−7.21 + 0.445i)14-s − 0.0542·15-s + (3.87 − 0.976i)16-s − 17-s + ⋯
L(s)  = 1  + (0.0616 + 0.998i)2-s + (0.0141 − 0.0141i)3-s + (−0.992 + 0.123i)4-s + (−0.496 − 0.496i)5-s + (0.0149 + 0.0132i)6-s + 1.93i·7-s + (−0.183 − 0.982i)8-s + 0.999i·9-s + (0.464 − 0.525i)10-s + (−0.939 − 0.939i)11-s + (−0.0122 + 0.0157i)12-s + (−0.283 + 0.283i)13-s + (−1.92 + 0.119i)14-s − 0.0140·15-s + (0.969 − 0.244i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.989 - 0.145i$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ -0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0562159 + 0.768870i\)
\(L(\frac12)\) \(\approx\) \(0.0562159 + 0.768870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 - 1.41i)T \)
17 \( 1 + T \)
good3 \( 1 + (-0.0244 + 0.0244i)T - 3iT^{2} \)
5 \( 1 + (1.10 + 1.10i)T + 5iT^{2} \)
7 \( 1 - 5.11iT - 7T^{2} \)
11 \( 1 + (3.11 + 3.11i)T + 11iT^{2} \)
13 \( 1 + (1.02 - 1.02i)T - 13iT^{2} \)
19 \( 1 + (4.22 - 4.22i)T - 19iT^{2} \)
23 \( 1 - 6.21iT - 23T^{2} \)
29 \( 1 + (-3.46 + 3.46i)T - 29iT^{2} \)
31 \( 1 - 7.78T + 31T^{2} \)
37 \( 1 + (-0.00459 - 0.00459i)T + 37iT^{2} \)
41 \( 1 + 1.98iT - 41T^{2} \)
43 \( 1 + (-6.99 - 6.99i)T + 43iT^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 + (-2.53 - 2.53i)T + 53iT^{2} \)
59 \( 1 + (-3.70 - 3.70i)T + 59iT^{2} \)
61 \( 1 + (0.657 - 0.657i)T - 61iT^{2} \)
67 \( 1 + (2.14 - 2.14i)T - 67iT^{2} \)
71 \( 1 + 3.45iT - 71T^{2} \)
73 \( 1 + 5.94iT - 73T^{2} \)
79 \( 1 + 6.42T + 79T^{2} \)
83 \( 1 + (5.30 - 5.30i)T - 83iT^{2} \)
89 \( 1 - 0.321iT - 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−12.45143013276284190536591201941, −11.68816397251876689526308571232, −10.31077917834112145200420846948, −9.062215661311859628547199845186, −8.308552535893205202596749697084, −7.84603289063236205904443183363, −6.10910312833581617036251952204, −5.48455719639142021689000080643, −4.45631124487443957694721292709, −2.61458635296188546981993159255, 0.57915631259869960641414961877, 2.73939564386504486457505227765, 4.00926376433305123857122276119, 4.69863492052918434492437939968, 6.70175391367916508398894819246, 7.50738470890159957953041076606, 8.720797341282131327602312189389, 10.09495269706530489231450972276, 10.45042529471062925314408023862, 11.27457532411250609633857309757

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