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Properties

Label 2-17e2-17.16-c3-0-3
Degree $2$
Conductor $289$
Sign $-0.743 - 0.669i$
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  − 1.26·2-s + 6.09i·3-s − 6.40·4-s − 13.2i·5-s − 7.69i·6-s − 27.2i·7-s + 18.1·8-s − 10.1·9-s + 16.7i·10-s + 45.5i·11-s − 39.0i·12-s − 52.5·13-s + 34.3i·14-s + 80.8·15-s + 28.3·16-s + ⋯
L(s)  = 1  − 0.446·2-s + 1.17i·3-s − 0.801·4-s − 1.18i·5-s − 0.523i·6-s − 1.47i·7-s + 0.803·8-s − 0.376·9-s + 0.529i·10-s + 1.24i·11-s − 0.939i·12-s − 1.12·13-s + 0.656i·14-s + 1.39·15-s + 0.442·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.743 - 0.669i$
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: \((2,\ 289,\ (\ :3/2),\ -0.743 - 0.669i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4495464113\)
\(L(\frac12)\) \(\approx\) \(0.4495464113\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.26T + 8T^{2} \)
3 \( 1 - 6.09iT - 27T^{2} \)
5 \( 1 + 13.2iT - 125T^{2} \)
7 \( 1 + 27.2iT - 343T^{2} \)
11 \( 1 - 45.5iT - 1.33e3T^{2} \)
13 \( 1 + 52.5T + 2.19e3T^{2} \)
19 \( 1 + 3.08T + 6.85e3T^{2} \)
23 \( 1 - 112. iT - 1.21e4T^{2} \)
29 \( 1 - 18.6iT - 2.43e4T^{2} \)
31 \( 1 + 238. iT - 2.97e4T^{2} \)
37 \( 1 - 162. iT - 5.06e4T^{2} \)
41 \( 1 - 383. iT - 6.89e4T^{2} \)
43 \( 1 + 468.T + 7.95e4T^{2} \)
47 \( 1 + 199.T + 1.03e5T^{2} \)
53 \( 1 - 105.T + 1.48e5T^{2} \)
59 \( 1 + 207.T + 2.05e5T^{2} \)
61 \( 1 - 586. iT - 2.26e5T^{2} \)
67 \( 1 - 401.T + 3.00e5T^{2} \)
71 \( 1 - 481. iT - 3.57e5T^{2} \)
73 \( 1 - 725. iT - 3.89e5T^{2} \)
79 \( 1 + 382. iT - 4.93e5T^{2} \)
83 \( 1 - 182.T + 5.71e5T^{2} \)
89 \( 1 + 623.T + 7.04e5T^{2} \)
97 \( 1 - 369. iT - 9.12e5T^{2} \)
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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

−11.57188303238395796815775765208, −10.14212682775083747772719421985, −9.923061367138876367674850286076, −9.244533977900731650372659680276, −8.030396018572013806660545993228, −7.20079099967040422101961068247, −5.02652469210750162428219646638, −4.63270104114502766550555162071, −3.86045335286467875478125840343, −1.27072078520152057268920980832, 0.22179341259285428448282684083, 2.05505031002207032230404752119, 3.16748071480551305263660571749, 5.11755515167635343386157290623, 6.24306673279536892245399648402, 7.13309416169899587217263878063, 8.201762168754265515917053525787, 8.865402887258501846148086576159, 10.02712225156530363025229597848, 10.99549376572885991085754820929

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