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Properties

Label 2-17e2-17.16-c3-0-52
Degree $2$
Conductor $289$
Sign $-0.242 - 0.970i$
Analytic cond. $17.0515$
Root an. cond. $4.12935$
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  − 4.67·2-s − 7.62i·3-s + 13.8·4-s − 11.9i·5-s + 35.6i·6-s − 26.1i·7-s − 27.1·8-s − 31.2·9-s + 55.6i·10-s + 3.24i·11-s − 105. i·12-s − 20.0·13-s + 122. i·14-s − 90.9·15-s + 16.4·16-s + ⋯
L(s)  = 1  − 1.65·2-s − 1.46i·3-s + 1.72·4-s − 1.06i·5-s + 2.42i·6-s − 1.41i·7-s − 1.20·8-s − 1.15·9-s + 1.76i·10-s + 0.0889i·11-s − 2.53i·12-s − 0.427·13-s + 2.32i·14-s − 1.56·15-s + 0.257·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(17.0515\)
Root analytic conductor: \(4.12935\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ -0.242 - 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5101027771\)
\(L(\frac12)\) \(\approx\) \(0.5101027771\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + 4.67T + 8T^{2} \)
3 \( 1 + 7.62iT - 27T^{2} \)
5 \( 1 + 11.9iT - 125T^{2} \)
7 \( 1 + 26.1iT - 343T^{2} \)
11 \( 1 - 3.24iT - 1.33e3T^{2} \)
13 \( 1 + 20.0T + 2.19e3T^{2} \)
19 \( 1 + 57.3T + 6.85e3T^{2} \)
23 \( 1 + 77.0iT - 1.21e4T^{2} \)
29 \( 1 + 286. iT - 2.43e4T^{2} \)
31 \( 1 + 8.54iT - 2.97e4T^{2} \)
37 \( 1 - 357. iT - 5.06e4T^{2} \)
41 \( 1 + 194. iT - 6.89e4T^{2} \)
43 \( 1 - 74.2T + 7.95e4T^{2} \)
47 \( 1 - 23.6T + 1.03e5T^{2} \)
53 \( 1 + 104.T + 1.48e5T^{2} \)
59 \( 1 + 249.T + 2.05e5T^{2} \)
61 \( 1 - 370. iT - 2.26e5T^{2} \)
67 \( 1 - 939.T + 3.00e5T^{2} \)
71 \( 1 + 520. iT - 3.57e5T^{2} \)
73 \( 1 - 348. iT - 3.89e5T^{2} \)
79 \( 1 - 953. iT - 4.93e5T^{2} \)
83 \( 1 - 1.41e3T + 5.71e5T^{2} \)
89 \( 1 + 486.T + 7.04e5T^{2} \)
97 \( 1 + 685. iT - 9.12e5T^{2} \)
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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

−10.49263095122590981314332268951, −9.649533831375952882374731288594, −8.480361954454966950348339160169, −7.935704401451775043029376386220, −7.13892786294279327840566177410, −6.35758847782402755886763644468, −4.47805033746414679613024887313, −2.18621243480024553835765932173, −1.03955004032881238331046311877, −0.39623618305366621804978316780, 2.22191970294586192202854336081, 3.32703597479481057278785521371, 5.07468135228350377505467454958, 6.31719899123137951561442875224, 7.47249602750678720209408966121, 8.731909776945400901206542230555, 9.215405307656922275405904128391, 10.03799822700645278118598749717, 10.82142722311810076253537594310, 11.29686105332932503206560412335

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