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Properties

Label 1-87-87.14-r0-0-0
Degree $1$
Conductor $87$
Sign $0.653 - 0.757i$
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  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.781 + 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.781 + 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.653 - 0.757i$
Analytic conductor: \(0.404026\)
Root analytic conductor: \(0.404026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (0:\ ),\ 0.653 - 0.757i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421755791 - 0.6513343394i\)
\(L(\frac12)\) \(\approx\) \(1.421755791 - 0.6513343394i\)
\(L(1)\) \(\approx\) \(1.515852990 - 0.4500496064i\)
\(L(1)\) \(\approx\) \(1.515852990 - 0.4500496064i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.974 - 0.222i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (0.781 + 0.623i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.433 + 0.900i)T \)
23 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + (-0.974 + 0.222i)T \)
37 \( 1 + (-0.781 + 0.623i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.433 + 0.900i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.974 - 0.222i)T \)
79 \( 1 + (0.781 - 0.623i)T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (0.974 - 0.222i)T \)
97 \( 1 + (-0.433 - 0.900i)T \)
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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

−30.89026172398750874494775550922, −29.742793875657619518929915012241, −29.25903499174205135062456765449, −27.519076565622611064780442134231, −26.31082661598456412128344109357, −25.35119678428309843541619662355, −24.431029211545119037423290038712, −23.03243733098151225684881066217, −22.32488662400290816585896809259, −21.67675365073721216477205350048, −19.99771609141464441824491696895, −19.19911887378523928454212084481, −17.68864723266802555300610930589, −16.218509719605149305172125816746, −15.3569082334792576304779636973, −14.31837270470587908452909627881, −13.23859370399916758783451173633, −11.953040253261293724822240752341, −11.012147489921689583595193797597, −9.44966914895003161058044106723, −7.52102882701628915212348136024, −6.574743851215106038782826367061, −5.38185497151867450100838436564, −3.54527762116227231774438990762, −2.71411951507047086895996879193, 1.680058947455952566698098077860, 3.66492229145882049030833676956, 4.608553182447082972319010892505, 6.12615541320523178926089779469, 7.33442938340726552147533383001, 9.19707778446097620071014503596, 10.4067920096107605338908308694, 12.09772707907527712811988299711, 12.577531439107180311650838311023, 13.84644044675218120637247718965, 14.9944223897229628148512182522, 16.34269154690480482507510727691, 16.94900454072427332517522125785, 19.16279486027204184152134600077, 19.91965201615822405811827190441, 20.82540970352698969528802332031, 22.08530589564730710724929133546, 22.99824430664483791647103225897, 24.033397634976995751827616436969, 24.894037138004055213298481323429, 26.04063018964394281476730281887, 27.61958631095917095647895629853, 28.74398701160786533537529161645, 29.363587569558026691042331963509, 30.71240359934784811776354312433

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