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Properties

Label 2-272-16.5-c1-0-11
Degree $2$
Conductor $272$
Sign $0.955 - 0.295i$
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  + (−0.812 + 1.15i)2-s + (−1.22 + 1.22i)3-s + (−0.678 − 1.88i)4-s + (−2.77 − 2.77i)5-s + (−0.423 − 2.42i)6-s + 0.669i·7-s + (2.72 + 0.743i)8-s − 0.0256i·9-s + (5.46 − 0.954i)10-s + (2.63 + 2.63i)11-s + (3.14 + 1.47i)12-s + (4.59 − 4.59i)13-s + (−0.775 − 0.544i)14-s + 6.81·15-s + (−3.07 + 2.55i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.574 + 0.818i)2-s + (−0.710 + 0.710i)3-s + (−0.339 − 0.940i)4-s + (−1.23 − 1.23i)5-s + (−0.172 − 0.989i)6-s + 0.253i·7-s + (0.964 + 0.262i)8-s − 0.00856i·9-s + (1.72 − 0.301i)10-s + (0.793 + 0.793i)11-s + (0.908 + 0.427i)12-s + (1.27 − 1.27i)13-s + (−0.207 − 0.145i)14-s + 1.76·15-s + (−0.769 + 0.638i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $0.955 - 0.295i$
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: \((2,\ 272,\ (\ :1/2),\ 0.955 - 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.590642 + 0.0891886i\)
\(L(\frac12)\) \(\approx\) \(0.590642 + 0.0891886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.812 - 1.15i)T \)
17 \( 1 + T \)
good3 \( 1 + (1.22 - 1.22i)T - 3iT^{2} \)
5 \( 1 + (2.77 + 2.77i)T + 5iT^{2} \)
7 \( 1 - 0.669iT - 7T^{2} \)
11 \( 1 + (-2.63 - 2.63i)T + 11iT^{2} \)
13 \( 1 + (-4.59 + 4.59i)T - 13iT^{2} \)
19 \( 1 + (-4.22 + 4.22i)T - 19iT^{2} \)
23 \( 1 + 2.17iT - 23T^{2} \)
29 \( 1 + (-1.35 + 1.35i)T - 29iT^{2} \)
31 \( 1 - 7.26T + 31T^{2} \)
37 \( 1 + (4.69 + 4.69i)T + 37iT^{2} \)
41 \( 1 + 2.91iT - 41T^{2} \)
43 \( 1 + (-0.0731 - 0.0731i)T + 43iT^{2} \)
47 \( 1 - 4.07T + 47T^{2} \)
53 \( 1 + (2.97 + 2.97i)T + 53iT^{2} \)
59 \( 1 + (-2.37 - 2.37i)T + 59iT^{2} \)
61 \( 1 + (1.26 - 1.26i)T - 61iT^{2} \)
67 \( 1 + (-9.07 + 9.07i)T - 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 - 4.88iT - 73T^{2} \)
79 \( 1 + 7.21T + 79T^{2} \)
83 \( 1 + (9.30 - 9.30i)T - 83iT^{2} \)
89 \( 1 + 16.9iT - 89T^{2} \)
97 \( 1 - 5.46T + 97T^{2} \)
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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

−11.73686805089783686733229116263, −10.97763692000685839189968250710, −9.934419297728229102738153486888, −8.869567859962839925054407867993, −8.247127581021898025003706539684, −7.18096933950210089645742227211, −5.74873578323977865586115226074, −4.89672073943864927236342698253, −4.07887080211226610009751181844, −0.78473953631294649242790483065, 1.20175424886667575150993863925, 3.32573058756533298207593985982, 4.01238152998661626592338349070, 6.31757290470528314062607779158, 6.96718997276725102145815480938, 7.955309055801406763365850183246, 8.975917820131612087481369273898, 10.32590119522356108687079637134, 11.32506147077702701791816165370, 11.58765095084489855425942787139

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