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Properties

Label 1-17e2-289.135-r0-0-0
Degree $1$
Conductor $289$
Sign $0.677 + 0.735i$
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  + (0.932 + 0.361i)2-s + (0.602 − 0.798i)3-s + (0.739 + 0.673i)4-s + (−0.445 + 0.895i)5-s + (0.850 − 0.526i)6-s + (0.273 + 0.961i)7-s + (0.445 + 0.895i)8-s + (−0.273 − 0.961i)9-s + (−0.739 + 0.673i)10-s + (−0.739 − 0.673i)11-s + (0.982 − 0.183i)12-s + (0.445 + 0.895i)13-s + (−0.0922 + 0.995i)14-s + (0.445 + 0.895i)15-s + (0.0922 + 0.995i)16-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)2-s + (0.602 − 0.798i)3-s + (0.739 + 0.673i)4-s + (−0.445 + 0.895i)5-s + (0.850 − 0.526i)6-s + (0.273 + 0.961i)7-s + (0.445 + 0.895i)8-s + (−0.273 − 0.961i)9-s + (−0.739 + 0.673i)10-s + (−0.739 − 0.673i)11-s + (0.982 − 0.183i)12-s + (0.445 + 0.895i)13-s + (−0.0922 + 0.995i)14-s + (0.445 + 0.895i)15-s + (0.0922 + 0.995i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(1.34211\)
Root analytic conductor: \(1.34211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (0:\ ),\ 0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.244955571 + 0.9839631366i\)
\(L(\frac12)\) \(\approx\) \(2.244955571 + 0.9839631366i\)
\(L(1)\) \(\approx\) \(1.918488161 + 0.4818068920i\)
\(L(1)\) \(\approx\) \(1.918488161 + 0.4818068920i\)

Euler product

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

−25.16177883768549271976052099312, −24.45604452165912982559257475444, −23.29314238646635768464129482923, −22.85657625731900638693614549626, −21.58314006902481934012537066340, −20.60722964232985042654125669973, −20.335960601826953905084574862275, −19.716380140802550655615360996326, −18.21346018205570801129246547249, −16.634731987950921302192997632693, −16.06559965523229208547594955709, −15.13637081130309792567598018641, −14.312280200059526433837876308422, −13.24030823951872442371324711859, −12.655861665103509949367027275144, −11.23546366635668694074203908067, −10.47579895939509105110422806336, −9.56460179874619914574416124420, −8.14319607390146340954844994430, −7.34185706208955951305299603082, −5.47837507434336335738042662943, −4.72699453898075311839067949092, −3.90705273065814508830332670536, −2.89103599762962914037329308324, −1.30011269308577804451448483467, 2.041061463103117237580243040469, 2.91149747392829393704749744526, 3.82997168750277836265215727573, 5.50384399325332822724094260502, 6.34727796766074895530926314660, 7.435479700898946961823186358140, 8.07304379194655388075136808269, 9.284307790290711107758250745716, 11.31457749869232061781822436943, 11.55545303528582698797430724482, 12.82480939149629886070165264108, 13.71301388073699634057209766164, 14.43452987936758720994870268253, 15.34352849019463114242875292155, 15.96727826468522624952723548931, 17.52706899911783076734468250044, 18.59463192547158604199014949865, 19.067356233911648402310184555174, 20.33601430746206808398757142189, 21.27841980723620428108175267666, 22.05912966092328913128598911682, 23.09796084673557071462120871103, 23.97479241305466393695280794527, 24.425872201256535232926993671080, 25.65566132791231470231126382609

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