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Properties

Label 2-17e2-17.16-c3-0-43
Degree $2$
Conductor $289$
Sign $0.911 - 0.410i$
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  + 5.35·2-s + 1.66i·3-s + 20.6·4-s − 5.96i·5-s + 8.89i·6-s + 27.8i·7-s + 67.6·8-s + 24.2·9-s − 31.9i·10-s − 18.6i·11-s + 34.3i·12-s − 42.5·13-s + 148. i·14-s + 9.92·15-s + 197.·16-s + ⋯
L(s)  = 1  + 1.89·2-s + 0.319i·3-s + 2.58·4-s − 0.533i·5-s + 0.605i·6-s + 1.50i·7-s + 2.99·8-s + 0.897·9-s − 1.01i·10-s − 0.510i·11-s + 0.825i·12-s − 0.907·13-s + 2.84i·14-s + 0.170·15-s + 3.07·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.911 - 0.410i$
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: \((2,\ 289,\ (\ :3/2),\ 0.911 - 0.410i)\)

Particular Values

\(L(2)\) \(\approx\) \(6.082949135\)
\(L(\frac12)\) \(\approx\) \(6.082949135\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.35T + 8T^{2} \)
3 \( 1 - 1.66iT - 27T^{2} \)
5 \( 1 + 5.96iT - 125T^{2} \)
7 \( 1 - 27.8iT - 343T^{2} \)
11 \( 1 + 18.6iT - 1.33e3T^{2} \)
13 \( 1 + 42.5T + 2.19e3T^{2} \)
19 \( 1 + 31.3T + 6.85e3T^{2} \)
23 \( 1 + 60.3iT - 1.21e4T^{2} \)
29 \( 1 - 117. iT - 2.43e4T^{2} \)
31 \( 1 + 228. iT - 2.97e4T^{2} \)
37 \( 1 - 99.4iT - 5.06e4T^{2} \)
41 \( 1 + 270. iT - 6.89e4T^{2} \)
43 \( 1 + 108.T + 7.95e4T^{2} \)
47 \( 1 + 250.T + 1.03e5T^{2} \)
53 \( 1 + 294.T + 1.48e5T^{2} \)
59 \( 1 + 62.0T + 2.05e5T^{2} \)
61 \( 1 + 799. iT - 2.26e5T^{2} \)
67 \( 1 + 645.T + 3.00e5T^{2} \)
71 \( 1 - 1.14e3iT - 3.57e5T^{2} \)
73 \( 1 + 550. iT - 3.89e5T^{2} \)
79 \( 1 - 253. iT - 4.93e5T^{2} \)
83 \( 1 + 717.T + 5.71e5T^{2} \)
89 \( 1 + 1.59e3T + 7.04e5T^{2} \)
97 \( 1 - 255. iT - 9.12e5T^{2} \)
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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

−11.84407636680414696977592008119, −10.92295311122068026778565063060, −9.710227879755166665316460630353, −8.468086337971800717573082856995, −7.09587843649677502514314628737, −6.03762428741794652408179499464, −5.13863293338772288053554672899, −4.46027042360068683027180402557, −3.10095200152947351947591415955, −1.97756211765876270537495018452, 1.59241975525837217561504356150, 3.03896090414133088356855388100, 4.19715463414717376442211200495, 4.84751415585457325362701182020, 6.40468725210675726619979451519, 7.12468834812939722923441857571, 7.58480894181597684280607970305, 10.01911124353441603614195314082, 10.59153516113960763477038538467, 11.59434898500472526708432239363

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