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Properties

Label 1-17e2-289.6-r1-0-0
Degree $1$
Conductor $289$
Sign $-0.972 + 0.232i$
Analytic cond. $31.0573$
Root an. cond. $31.0573$
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  + (−0.873 + 0.486i)2-s + (−0.295 − 0.955i)3-s + (0.526 − 0.850i)4-s + (0.339 − 0.940i)5-s + (0.723 + 0.690i)6-s + (0.545 − 0.837i)7-s + (−0.0461 + 0.998i)8-s + (−0.824 + 0.565i)9-s + (0.160 + 0.986i)10-s + (0.584 + 0.811i)11-s + (−0.967 − 0.251i)12-s + (−0.673 − 0.739i)13-s + (−0.0692 + 0.997i)14-s + (−0.998 − 0.0461i)15-s + (−0.445 − 0.895i)16-s + ⋯
L(s)  = 1  + (−0.873 + 0.486i)2-s + (−0.295 − 0.955i)3-s + (0.526 − 0.850i)4-s + (0.339 − 0.940i)5-s + (0.723 + 0.690i)6-s + (0.545 − 0.837i)7-s + (−0.0461 + 0.998i)8-s + (−0.824 + 0.565i)9-s + (0.160 + 0.986i)10-s + (0.584 + 0.811i)11-s + (−0.967 − 0.251i)12-s + (−0.673 − 0.739i)13-s + (−0.0692 + 0.997i)14-s + (−0.998 − 0.0461i)15-s + (−0.445 − 0.895i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.972 + 0.232i$
Analytic conductor: \(31.0573\)
Root analytic conductor: \(31.0573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 289,\ (1:\ ),\ -0.972 + 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08137867544 - 0.6891694910i\)
\(L(\frac12)\) \(\approx\) \(-0.08137867544 - 0.6891694910i\)
\(L(1)\) \(\approx\) \(0.5680129617 - 0.3271664384i\)
\(L(1)\) \(\approx\) \(0.5680129617 - 0.3271664384i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (-0.873 + 0.486i)T \)
3 \( 1 + (-0.295 - 0.955i)T \)
5 \( 1 + (0.339 - 0.940i)T \)
7 \( 1 + (0.545 - 0.837i)T \)
11 \( 1 + (0.584 + 0.811i)T \)
13 \( 1 + (-0.673 - 0.739i)T \)
19 \( 1 + (0.486 - 0.873i)T \)
23 \( 1 + (-0.837 - 0.545i)T \)
29 \( 1 + (0.160 - 0.986i)T \)
31 \( 1 + (-0.424 + 0.905i)T \)
37 \( 1 + (-0.862 - 0.506i)T \)
41 \( 1 + (0.884 + 0.466i)T \)
43 \( 1 + (-0.317 - 0.948i)T \)
47 \( 1 + (0.183 + 0.982i)T \)
53 \( 1 + (-0.565 - 0.824i)T \)
59 \( 1 + (-0.403 + 0.914i)T \)
61 \( 1 + (0.690 - 0.723i)T \)
67 \( 1 + (0.273 + 0.961i)T \)
71 \( 1 + (-0.978 + 0.206i)T \)
73 \( 1 + (0.656 - 0.754i)T \)
79 \( 1 + (-0.115 + 0.993i)T \)
83 \( 1 + (0.638 - 0.769i)T \)
89 \( 1 + (-0.673 + 0.739i)T \)
97 \( 1 + (-0.206 - 0.978i)T \)
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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

−26.09621687460256068090895061542, −25.18100670287767951147371847582, −24.15107367408703810722036015881, −22.54537969042708472039439366699, −21.78723842552304540008399241785, −21.50571630133286994568397965689, −20.39851952391128011607795410758, −19.260879377803908127646836239815, −18.466277034388304484194981236056, −17.62772255342408501791682931259, −16.73988503222328174317041050210, −15.86656726813339549497890963721, −14.79401490973588322525120195052, −14.02066722159887113584653830112, −12.10608880316835676942511952619, −11.527614648823180584409029553429, −10.72176867314955304522951078297, −9.71256137929630351674246154081, −9.067648000531098859324753101162, −7.90731860612091296430585094263, −6.534339663516642337892727813809, −5.54786267215285117248077627435, −3.92794820914102066907694375062, −2.95242997204696338314413792619, −1.744683267198160780072378947615, 0.30529282301326464857909979910, 1.2445160054173820101159685347, 2.23982951391673860792010326102, 4.64259122758070127893075482083, 5.55057057929484883184891609291, 6.77534978016267957314492435625, 7.575654933584327426604753966035, 8.38734623702484085116095738840, 9.52329572807372047204043595182, 10.52759683361729167762029795229, 11.70348457158901604636438492566, 12.58519942798736120127134206888, 13.76573679872340279172421436741, 14.548699741494908747001577287897, 15.91505210388281141564649131730, 16.93937509254990806999662568542, 17.5966927256392657753137112505, 17.89115193810815371970735116622, 19.45336183492740730184994500873, 19.991717507006126929899935186493, 20.69222434914802384623188434162, 22.38030027509131604624118051066, 23.404876407071881507831944234292, 24.19948342514666073285983579399, 24.73411712514812008468236983143

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