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Properties

Label 2-272-17.13-c3-0-12
Degree $2$
Conductor $272$
Sign $-0.0767 - 0.997i$
Analytic cond. $16.0485$
Root an. cond. $4.00606$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.91 + 6.91i)3-s + (8.19 + 8.19i)5-s + (18.1 − 18.1i)7-s + 68.7i·9-s + (−44.7 + 44.7i)11-s + 9.95·13-s + 113. i·15-s + (51.7 − 47.2i)17-s − 108. i·19-s + 250.·21-s + (4.64 − 4.64i)23-s + 9.47i·25-s + (−288. + 288. i)27-s + (140. + 140. i)29-s + (−158. − 158. i)31-s + ⋯
L(s)  = 1  + (1.33 + 1.33i)3-s + (0.733 + 0.733i)5-s + (0.977 − 0.977i)7-s + 2.54i·9-s + (−1.22 + 1.22i)11-s + 0.212·13-s + 1.95i·15-s + (0.738 − 0.674i)17-s − 1.30i·19-s + 2.60·21-s + (0.0421 − 0.0421i)23-s + 0.0758i·25-s + (−2.05 + 2.05i)27-s + (0.900 + 0.900i)29-s + (−0.917 − 0.917i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0767 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0767 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $-0.0767 - 0.997i$
Analytic conductor: \(16.0485\)
Root analytic conductor: \(4.00606\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :3/2),\ -0.0767 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.358703749\)
\(L(\frac12)\) \(\approx\) \(3.358703749\)
\(L(\frac{5}{2})\) 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 \( 1 + (-51.7 + 47.2i)T \)
good3 \( 1 + (-6.91 - 6.91i)T + 27iT^{2} \)
5 \( 1 + (-8.19 - 8.19i)T + 125iT^{2} \)
7 \( 1 + (-18.1 + 18.1i)T - 343iT^{2} \)
11 \( 1 + (44.7 - 44.7i)T - 1.33e3iT^{2} \)
13 \( 1 - 9.95T + 2.19e3T^{2} \)
19 \( 1 + 108. iT - 6.85e3T^{2} \)
23 \( 1 + (-4.64 + 4.64i)T - 1.21e4iT^{2} \)
29 \( 1 + (-140. - 140. i)T + 2.43e4iT^{2} \)
31 \( 1 + (158. + 158. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-33.6 - 33.6i)T + 5.06e4iT^{2} \)
41 \( 1 + (55.7 - 55.7i)T - 6.89e4iT^{2} \)
43 \( 1 + 193. iT - 7.95e4T^{2} \)
47 \( 1 + 246.T + 1.03e5T^{2} \)
53 \( 1 - 208. iT - 1.48e5T^{2} \)
59 \( 1 - 163. iT - 2.05e5T^{2} \)
61 \( 1 + (-76.2 + 76.2i)T - 2.26e5iT^{2} \)
67 \( 1 - 292.T + 3.00e5T^{2} \)
71 \( 1 + (484. + 484. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-90.9 - 90.9i)T + 3.89e5iT^{2} \)
79 \( 1 + (-45.3 + 45.3i)T - 4.93e5iT^{2} \)
83 \( 1 + 661. iT - 5.71e5T^{2} \)
89 \( 1 + 657.T + 7.04e5T^{2} \)
97 \( 1 + (-58.0 - 58.0i)T + 9.12e5iT^{2} \)
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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

−11.20615379136856791620605676887, −10.38134104672347442632167731519, −10.03901325588118120323240748238, −9.018120977213043816024625189537, −7.84900943699355195470016045960, −7.20755603970983300961469253931, −5.11895742495284292006644959924, −4.48945148772242603723363482373, −3.07641947629624856031399300256, −2.12976259736680185991923621720, 1.24495164987727443966856156708, 2.13303666089833661088594040095, 3.31923602637938654832445391956, 5.39127655282429421788522930694, 6.11391432213144333366125282654, 7.77524726316661636203934178356, 8.329453317948534094326585875763, 8.796017768101529977649490621565, 10.02879085065620710784648667321, 11.51514647195518819951261681449

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