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Properties

Label 1-17e2-289.81-r0-0-0
Degree $1$
Conductor $289$
Sign $0.987 + 0.156i$
Analytic cond. $1.34211$
Root an. cond. $1.34211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.273 − 0.961i)2-s + (0.995 + 0.0922i)3-s + (−0.850 − 0.526i)4-s + (−0.673 + 0.739i)5-s + (0.361 − 0.932i)6-s + (−0.183 + 0.982i)7-s + (−0.739 + 0.673i)8-s + (0.982 + 0.183i)9-s + (0.526 + 0.850i)10-s + (−0.526 + 0.850i)11-s + (−0.798 − 0.602i)12-s + (0.739 − 0.673i)13-s + (0.895 + 0.445i)14-s + (−0.739 + 0.673i)15-s + (0.445 + 0.895i)16-s + ⋯
L(s)  = 1  + (0.273 − 0.961i)2-s + (0.995 + 0.0922i)3-s + (−0.850 − 0.526i)4-s + (−0.673 + 0.739i)5-s + (0.361 − 0.932i)6-s + (−0.183 + 0.982i)7-s + (−0.739 + 0.673i)8-s + (0.982 + 0.183i)9-s + (0.526 + 0.850i)10-s + (−0.526 + 0.850i)11-s + (−0.798 − 0.602i)12-s + (0.739 − 0.673i)13-s + (0.895 + 0.445i)14-s + (−0.739 + 0.673i)15-s + (0.445 + 0.895i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.987 + 0.156i$
Analytic conductor: \(1.34211\)
Root analytic conductor: \(1.34211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (0:\ ),\ 0.987 + 0.156i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509395648 + 0.1192047629i\)
\(L(\frac12)\) \(\approx\) \(1.509395648 + 0.1192047629i\)
\(L(1)\) \(\approx\) \(1.307627653 - 0.1720137102i\)
\(L(1)\) \(\approx\) \(1.307627653 - 0.1720137102i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (0.273 - 0.961i)T \)
3 \( 1 + (0.995 + 0.0922i)T \)
5 \( 1 + (-0.673 + 0.739i)T \)
7 \( 1 + (-0.183 + 0.982i)T \)
11 \( 1 + (-0.526 + 0.850i)T \)
13 \( 1 + (0.739 - 0.673i)T \)
19 \( 1 + (0.273 + 0.961i)T \)
23 \( 1 + (-0.183 + 0.982i)T \)
29 \( 1 + (0.526 - 0.850i)T \)
31 \( 1 + (0.673 + 0.739i)T \)
37 \( 1 + (0.798 - 0.602i)T \)
41 \( 1 + (-0.995 - 0.0922i)T \)
43 \( 1 + (-0.445 + 0.895i)T \)
47 \( 1 + (-0.982 + 0.183i)T \)
53 \( 1 + (0.982 + 0.183i)T \)
59 \( 1 + (-0.932 + 0.361i)T \)
61 \( 1 + (0.361 - 0.932i)T \)
67 \( 1 + (-0.273 - 0.961i)T \)
71 \( 1 + (0.183 - 0.982i)T \)
73 \( 1 + (-0.895 + 0.445i)T \)
79 \( 1 + (-0.961 + 0.273i)T \)
83 \( 1 + (-0.0922 - 0.995i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (0.183 - 0.982i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.50794292836883447561826885006, −24.36178302763150614589751844180, −23.92143527796844114588285472569, −23.24292558628371716279289735560, −21.87056956451179667770369836635, −20.89501172134557155945973938486, −20.122405119261560940187361369714, −19.09200819062694337530020860577, −18.26069071027399228236180140208, −16.83190697056653050197093286274, −16.18686028894481382987251822690, −15.48252566766975976569538537386, −14.35111632185704845945648668123, −13.42115831704195758889400221774, −13.09565554107433117478333407044, −11.660123846063695200585711836327, −10.138434622987711170360039592200, −8.83383764765054701411815361885, −8.371199553610708537422387392228, −7.38563402457848661132588498938, −6.487281481874303538361922266418, −4.83902337011953037008970942767, −4.038135884402618268842182384729, −3.107705039186638930942987902901, −0.890536813160784779827596542642, 1.77992179138706837787921114038, 2.87483122800175802489025065393, 3.51308673081233174572975414642, 4.72712024382070375279409425184, 6.1123820510406035837905920061, 7.73360917351088006244572957884, 8.47709617055981769772559056397, 9.71657042582919567023670757642, 10.356235767527612134921345692008, 11.61167722683932450321553086545, 12.47206574875181299079931773752, 13.38486156095074350173065558544, 14.43694569856981363132575688050, 15.25952004600916257147262185326, 15.771529906378368141771131232638, 18.02193938796939468792029466020, 18.424777018875530078699610626467, 19.36657954705659171337007763234, 20.01970175366355900348029897439, 21.004562340717270951657962162315, 21.68635117880675220221811260907, 22.81624748409085830365684484886, 23.32908493011252070485752636592, 24.74983618528837176160316976665, 25.6246895854936078335278137699

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