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Properties

Label 2-17e2-17.16-c3-0-57
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  + 3.73·2-s − 3.65i·3-s + 5.93·4-s − 14.5i·5-s − 13.6i·6-s − 12.9i·7-s − 7.71·8-s + 13.6·9-s − 54.1i·10-s + 20.2i·11-s − 21.6i·12-s − 90.7·13-s − 48.4i·14-s − 53.0·15-s − 76.2·16-s + ⋯
L(s)  = 1  + 1.31·2-s − 0.703i·3-s + 0.741·4-s − 1.29i·5-s − 0.928i·6-s − 0.700i·7-s − 0.340·8-s + 0.505·9-s − 1.71i·10-s + 0.555i·11-s − 0.521i·12-s − 1.93·13-s − 0.924i·14-s − 0.913·15-s − 1.19·16-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

\(L(2)\) \(\approx\) \(2.881903110\)
\(L(\frac12)\) \(\approx\) \(2.881903110\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.73T + 8T^{2} \)
3 \( 1 + 3.65iT - 27T^{2} \)
5 \( 1 + 14.5iT - 125T^{2} \)
7 \( 1 + 12.9iT - 343T^{2} \)
11 \( 1 - 20.2iT - 1.33e3T^{2} \)
13 \( 1 + 90.7T + 2.19e3T^{2} \)
19 \( 1 - 127.T + 6.85e3T^{2} \)
23 \( 1 - 69.5iT - 1.21e4T^{2} \)
29 \( 1 + 43.9iT - 2.43e4T^{2} \)
31 \( 1 + 218. iT - 2.97e4T^{2} \)
37 \( 1 + 41.5iT - 5.06e4T^{2} \)
41 \( 1 + 440. iT - 6.89e4T^{2} \)
43 \( 1 - 310.T + 7.95e4T^{2} \)
47 \( 1 - 84.3T + 1.03e5T^{2} \)
53 \( 1 + 47.6T + 1.48e5T^{2} \)
59 \( 1 - 1.73T + 2.05e5T^{2} \)
61 \( 1 + 159. iT - 2.26e5T^{2} \)
67 \( 1 - 141.T + 3.00e5T^{2} \)
71 \( 1 - 447. iT - 3.57e5T^{2} \)
73 \( 1 - 757. iT - 3.89e5T^{2} \)
79 \( 1 + 529. iT - 4.93e5T^{2} \)
83 \( 1 - 762.T + 5.71e5T^{2} \)
89 \( 1 + 397.T + 7.04e5T^{2} \)
97 \( 1 + 427. iT - 9.12e5T^{2} \)
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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

−11.67787473358403891787675282855, −9.939568322519734029275897456014, −9.251298889155858358259559808367, −7.60539149703587285412653877421, −7.15974151306418190650626971619, −5.60245631408225356905780721647, −4.81087124658601941299259982863, −4.00789884270428056069950666342, −2.25938018335810978385888494518, −0.69109735288501943773774985298, 2.64584266295084066816706732846, 3.29318925406742165476917644705, 4.64644738556275847335298867430, 5.40469766741640186975243946355, 6.59915513886318050729978911833, 7.47212261518916367131820141240, 9.178190730978938572664250019652, 9.984076270813754195278711592805, 10.93576182586203017832803682067, 11.93672867654383504463723035980

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