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Properties

Label 1-87-87.17-r0-0-0
Degree $1$
Conductor $87$
Sign $0.189 + 0.981i$
Analytic cond. $0.404026$
Root an. cond. $0.404026$
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  + i·2-s − 4-s + 5-s + 7-s − i·8-s + i·10-s + i·11-s − 13-s + i·14-s + 16-s + i·17-s − i·19-s − 20-s − 22-s − 23-s + ⋯
L(s)  = 1  + i·2-s − 4-s + 5-s + 7-s − i·8-s + i·10-s + i·11-s − 13-s + i·14-s + 16-s + i·17-s − i·19-s − 20-s − 22-s − 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.189 + 0.981i$
Analytic conductor: \(0.404026\)
Root analytic conductor: \(0.404026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (0:\ ),\ 0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8263990864 + 0.6824343858i\)
\(L(\frac12)\) \(\approx\) \(0.8263990864 + 0.6824343858i\)
\(L(1)\) \(\approx\) \(0.9703448320 + 0.5396066234i\)
\(L(1)\) \(\approx\) \(0.9703448320 + 0.5396066234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 \)
5 \( 1 + iT \)
7 \( 1 \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 \)
19 \( 1 + T \)
23 \( 1 - iT \)
31 \( 1 + iT \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 - T \)
47 \( 1 + iT \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 - T \)
79 \( 1 \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 \)
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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

−30.03162977003697425012088510749, −29.5313175256659592818029063068, −28.52617214559360284527703407277, −27.26857136285387503543170684852, −26.626470958888148881791754884387, −25.02140904357297021473776074364, −24.05271907756112473456260630062, −22.58306552387800776913489501730, −21.56416311279370417224326371103, −20.97036113291559205143280683908, −19.788817519962559863160112807009, −18.43633573987053271854543000759, −17.73746653769089972300807042781, −16.57232721063920334103144216794, −14.37961978362571003135622820542, −13.96780934143384557455688873936, −12.526553444680769451791861106930, −11.383459346550827576177227002601, −10.27366192306778053319536635190, −9.20239268407527426077041607097, −7.9146930989473827406403536926, −5.77248144528396589701120551281, −4.66972028835957293162577999341, −2.848112707153677537591047837683, −1.53206736323036325963787687077, 1.98448656213559888905659343027, 4.45253022610878389821673657839, 5.437437846052873020595572285158, 6.79928305512299886846284686132, 7.996713724909518969163014456257, 9.32506945777199325756335039158, 10.35778375356298836139000415790, 12.28575575137485412139777233105, 13.52743861307219557940951557654, 14.56241931544811906953688257822, 15.34005347922612509403309785956, 17.18418891512788774563795342083, 17.40908920817529469092148094322, 18.56123461001748083845084815741, 20.207600880352185510874303477542, 21.59823118327766315743187062521, 22.296168866809329884986831561506, 23.827499339857516975889764435006, 24.4778931895305754823999522706, 25.59402644430193894077754923411, 26.347470797402850567113411518407, 27.64683410715515917615520527634, 28.47563101901624910617416271080, 30.05156931051186456308008551727, 30.9369929476663766018384785265

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