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Properties

Label 2-17e2-17.16-c3-0-14
Degree $2$
Conductor $289$
Sign $-0.970 - 0.242i$
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  − 1.22·2-s + 0.534i·3-s − 6.49·4-s + 20.9i·5-s − 0.654i·6-s + 15.0i·7-s + 17.7·8-s + 26.7·9-s − 25.6i·10-s + 45.4i·11-s − 3.47i·12-s + 3.14·13-s − 18.4i·14-s − 11.2·15-s + 30.2·16-s + ⋯
L(s)  = 1  − 0.433·2-s + 0.102i·3-s − 0.812·4-s + 1.87i·5-s − 0.0445i·6-s + 0.811i·7-s + 0.784·8-s + 0.989·9-s − 0.811i·10-s + 1.24i·11-s − 0.0835i·12-s + 0.0670·13-s − 0.351i·14-s − 0.192·15-s + 0.472·16-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.970 - 0.242i$
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: \((2,\ 289,\ (\ :3/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.112488762\)
\(L(\frac12)\) \(\approx\) \(1.112488762\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.22T + 8T^{2} \)
3 \( 1 - 0.534iT - 27T^{2} \)
5 \( 1 - 20.9iT - 125T^{2} \)
7 \( 1 - 15.0iT - 343T^{2} \)
11 \( 1 - 45.4iT - 1.33e3T^{2} \)
13 \( 1 - 3.14T + 2.19e3T^{2} \)
19 \( 1 - 63.2T + 6.85e3T^{2} \)
23 \( 1 - 114. iT - 1.21e4T^{2} \)
29 \( 1 - 96.6iT - 2.43e4T^{2} \)
31 \( 1 + 194. iT - 2.97e4T^{2} \)
37 \( 1 + 73.6iT - 5.06e4T^{2} \)
41 \( 1 - 341. iT - 6.89e4T^{2} \)
43 \( 1 - 281.T + 7.95e4T^{2} \)
47 \( 1 - 36.2T + 1.03e5T^{2} \)
53 \( 1 + 191.T + 1.48e5T^{2} \)
59 \( 1 + 104.T + 2.05e5T^{2} \)
61 \( 1 + 517. iT - 2.26e5T^{2} \)
67 \( 1 + 560.T + 3.00e5T^{2} \)
71 \( 1 + 333. iT - 3.57e5T^{2} \)
73 \( 1 + 378. iT - 3.89e5T^{2} \)
79 \( 1 + 877. iT - 4.93e5T^{2} \)
83 \( 1 + 1.19e3T + 5.71e5T^{2} \)
89 \( 1 - 783.T + 7.04e5T^{2} \)
97 \( 1 + 1.60e3iT - 9.12e5T^{2} \)
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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

−11.63929297540625014764670441973, −10.61338639422973810184209676157, −9.816166143790924453745104560366, −9.345616002880608881676099414773, −7.67860960381086395926795830986, −7.24185079528261274129386314838, −5.94514369743079316558637833797, −4.55795547130807889991362634649, −3.33593914715865263050194524988, −1.88982299222477762321656796165, 0.60172015705340350101563296766, 1.21721915951622446063506611429, 3.91839022772439508424245548858, 4.64151041402721503144055869066, 5.66346528989209165613649658834, 7.35292015119128643153415638301, 8.323855610020554372219921040523, 8.908987458762955567270067470892, 9.808087124601730497942099727134, 10.68473024572181992190128323968

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