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Properties

Label 1-17e2-289.186-r0-0-0
Degree $1$
Conductor $289$
Sign $-0.156 - 0.987i$
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 + (−0.0922 − 0.995i)3-s + (−0.850 + 0.526i)4-s + (−0.739 + 0.673i)5-s + (−0.932 + 0.361i)6-s + (0.982 − 0.183i)7-s + (0.739 + 0.673i)8-s + (−0.982 + 0.183i)9-s + (0.850 + 0.526i)10-s + (0.850 − 0.526i)11-s + (0.602 + 0.798i)12-s + (0.739 + 0.673i)13-s + (−0.445 − 0.895i)14-s + (0.739 + 0.673i)15-s + (0.445 − 0.895i)16-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.0922 − 0.995i)3-s + (−0.850 + 0.526i)4-s + (−0.739 + 0.673i)5-s + (−0.932 + 0.361i)6-s + (0.982 − 0.183i)7-s + (0.739 + 0.673i)8-s + (−0.982 + 0.183i)9-s + (0.850 + 0.526i)10-s + (0.850 − 0.526i)11-s + (0.602 + 0.798i)12-s + (0.739 + 0.673i)13-s + (−0.445 − 0.895i)14-s + (0.739 + 0.673i)15-s + (0.445 − 0.895i)16-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.156 - 0.987i$
Analytic conductor: \(1.34211\)
Root analytic conductor: \(1.34211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (0:\ ),\ -0.156 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6302489987 - 0.7383329794i\)
\(L(\frac12)\) \(\approx\) \(0.6302489987 - 0.7383329794i\)
\(L(1)\) \(\approx\) \(0.7102314641 - 0.5123185736i\)
\(L(1)\) \(\approx\) \(0.7102314641 - 0.5123185736i\)

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 \( 1 + (-0.0922 - 0.995i)T \)
5 \( 1 + (-0.739 + 0.673i)T \)
7 \( 1 + (0.982 - 0.183i)T \)
11 \( 1 + (0.850 - 0.526i)T \)
13 \( 1 + (0.739 + 0.673i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.982 - 0.183i)T \)
29 \( 1 + (0.850 - 0.526i)T \)
31 \( 1 + (-0.739 - 0.673i)T \)
37 \( 1 + (0.602 - 0.798i)T \)
41 \( 1 + (-0.0922 - 0.995i)T \)
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 \( 1 + (-0.932 + 0.361i)T \)
67 \( 1 + (-0.273 + 0.961i)T \)
71 \( 1 + (0.982 - 0.183i)T \)
73 \( 1 + (-0.445 + 0.895i)T \)
79 \( 1 + (0.273 - 0.961i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (0.982 - 0.183i)T \)
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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

−25.59836979697767207581978694261, −25.03728936764981474545225002959, −23.87447433946857101541386656634, −23.261718496390550315176011114842, −22.32107863351736312285935341334, −21.30147962554333995601717550520, −20.25043407615163631973468482201, −19.546144595155424499632375792334, −18.07847739348730449564536221726, −17.30496339405469371930070523205, −16.55350849987195767775101913462, −15.526281304202880518766365517907, −15.09261816929962424459380371918, −14.17208134006416489560212446615, −12.83246887674588411739990183506, −11.51541453294161746539751118228, −10.69032056563573561320190566838, −9.30091643024074436299324580701, −8.69508721793775592714698552801, −7.86412466276706425035654766297, −6.51597823480934402598475352798, −5.07746640849539834851303016666, −4.71141119839789556428621989485, −3.544284083215186974681053576963, −1.1271731115665754728069916984, 1.00437048117220453245344901213, 2.074468076297196930145401320814, 3.40159551177601141947442257905, 4.39424701637074716409736739542, 6.063271859940205474591550400907, 7.305508921221083397583641914792, 8.17000396919384102157691285227, 8.95041482839841888854693517258, 10.65141896063701624109323120928, 11.394895958565343698351903474886, 11.82663935795394483431990398389, 12.99358629490236966141929149915, 14.14938573127431172790861845421, 14.557878860784396897513056558418, 16.44249665302192518291727466458, 17.36640839262344235992168306772, 18.2935204238332456841221448239, 18.93645982458798942407649909558, 19.55983752073266555587498239818, 20.59973882154762107975046501934, 21.52438974613418036283053685449, 22.70482876100732935646972994896, 23.280504605642879957023042620764, 24.17825867216745528308337199686, 25.294616240287181663660202115069

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