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Properties

Label 2-87-29.25-c1-0-5
Degree $2$
Conductor $87$
Sign $-0.543 + 0.839i$
Analytic cond. $0.694698$
Root an. cond. $0.833485$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.596 − 2.61i)2-s + (0.900 − 0.433i)3-s + (−4.67 − 2.25i)4-s + (−0.692 + 3.03i)5-s + (−0.596 − 2.61i)6-s + (2.13 − 1.02i)7-s + (−5.32 + 6.68i)8-s + (0.623 − 0.781i)9-s + (7.51 + 3.61i)10-s + (1.62 + 2.03i)11-s − 5.18·12-s + (−1.43 − 1.80i)13-s + (−1.41 − 6.19i)14-s + (0.692 + 3.03i)15-s + (7.81 + 9.80i)16-s − 4.83·17-s + ⋯
L(s)  = 1  + (0.421 − 1.84i)2-s + (0.520 − 0.250i)3-s + (−2.33 − 1.12i)4-s + (−0.309 + 1.35i)5-s + (−0.243 − 1.06i)6-s + (0.807 − 0.388i)7-s + (−1.88 + 2.36i)8-s + (0.207 − 0.260i)9-s + (2.37 + 1.14i)10-s + (0.489 + 0.613i)11-s − 1.49·12-s + (−0.399 − 0.500i)13-s + (−0.377 − 1.65i)14-s + (0.178 + 0.783i)15-s + (1.95 + 2.45i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.567605 - 1.04317i\)
\(L(\frac12)\) \(\approx\) \(0.567605 - 1.04317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-5.05 - 1.86i)T \)
good2 \( 1 + (-0.596 + 2.61i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (0.692 - 3.03i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-2.13 + 1.02i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-1.62 - 2.03i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.43 + 1.80i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 4.83T + 17T^{2} \)
19 \( 1 + (1.47 + 0.710i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.263 - 1.15i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (1.28 - 5.63i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-4.30 + 5.39i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 7.66T + 41T^{2} \)
43 \( 1 + (0.419 + 1.84i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (5.79 + 7.27i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.15 + 5.04i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 4.90T + 59T^{2} \)
61 \( 1 + (-10.3 + 5.00i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (4.34 - 5.45i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-2.32 - 2.92i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.532 + 2.33i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-2.34 + 2.93i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-9.54 - 4.59i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-1.26 + 5.55i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (15.1 + 7.28i)T + (60.4 + 75.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.73437766346699013638322099364, −12.62707780425592769190633463340, −11.53627588225444625585037759735, −10.80094571347772997344294808152, −9.937692601084918336223512194282, −8.547074224549717038125896281475, −6.95977393484443984075993531816, −4.68060400702176963914707979818, −3.39878355710959014789584512321, −2.09730961705240444748345141327, 4.27790468720207800230934456506, 4.94289481757606924129502933645, 6.40708763929686860673538073046, 7.989095101302902548308098827481, 8.576070669138318105499200333447, 9.335428144864701899625525865974, 11.75114440425857549769520759626, 12.96174536832155259188761446268, 13.79229880352355306982404814772, 14.77663531250962499054413475817

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