Properties

Label 2-17e2-17.16-c3-0-59
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  + 2.16·2-s − 9.14i·3-s − 3.31·4-s − 16.2i·5-s − 19.7i·6-s + 12.2i·7-s − 24.4·8-s − 56.5·9-s − 35.1i·10-s − 11.1i·11-s + 30.2i·12-s + 28.8·13-s + 26.4i·14-s − 148.·15-s − 26.5·16-s + ⋯
L(s)  = 1  + 0.765·2-s − 1.75i·3-s − 0.414·4-s − 1.45i·5-s − 1.34i·6-s + 0.660i·7-s − 1.08·8-s − 2.09·9-s − 1.11i·10-s − 0.306i·11-s + 0.728i·12-s + 0.616·13-s + 0.505i·14-s − 2.55·15-s − 0.414·16-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

\(L(2)\) \(\approx\) \(1.378961095\)
\(L(\frac12)\) \(\approx\) \(1.378961095\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.16T + 8T^{2} \)
3 \( 1 + 9.14iT - 27T^{2} \)
5 \( 1 + 16.2iT - 125T^{2} \)
7 \( 1 - 12.2iT - 343T^{2} \)
11 \( 1 + 11.1iT - 1.33e3T^{2} \)
13 \( 1 - 28.8T + 2.19e3T^{2} \)
19 \( 1 - 79.5T + 6.85e3T^{2} \)
23 \( 1 + 45.0iT - 1.21e4T^{2} \)
29 \( 1 + 20.3iT - 2.43e4T^{2} \)
31 \( 1 + 1.03iT - 2.97e4T^{2} \)
37 \( 1 - 219. iT - 5.06e4T^{2} \)
41 \( 1 + 310. iT - 6.89e4T^{2} \)
43 \( 1 + 483.T + 7.95e4T^{2} \)
47 \( 1 + 632.T + 1.03e5T^{2} \)
53 \( 1 + 490.T + 1.48e5T^{2} \)
59 \( 1 + 147.T + 2.05e5T^{2} \)
61 \( 1 - 176. iT - 2.26e5T^{2} \)
67 \( 1 - 809.T + 3.00e5T^{2} \)
71 \( 1 + 714. iT - 3.57e5T^{2} \)
73 \( 1 + 780. iT - 3.89e5T^{2} \)
79 \( 1 + 230. iT - 4.93e5T^{2} \)
83 \( 1 + 236.T + 5.71e5T^{2} \)
89 \( 1 + 688.T + 7.04e5T^{2} \)
97 \( 1 - 1.84e3iT - 9.12e5T^{2} \)
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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

−11.50857598703378261830423380220, −9.444801678737598177923252246174, −8.550056475476659541305008069858, −8.105767562998550242903882052523, −6.56452977717525545928998296871, −5.67442642944717666334700563076, −4.90145269071228706585964032052, −3.21024472956880208296038514216, −1.59609083206107733879308540294, −0.41658086189744174866980937577, 3.12311894574710407737420309321, 3.61797148507357637736639647034, 4.63183875704019633513199397162, 5.63355801817640506876540145525, 6.77936882189359794895437429859, 8.266216015552184395679651798138, 9.603068755772873248942021155877, 9.981316241088659678142204031070, 11.03518553996724610392000358174, 11.53535099553979226936222553776

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