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Properties

Label 2-87-29.7-c1-0-2
Degree $2$
Conductor $87$
Sign $0.773 - 0.633i$
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  + (0.203 + 0.892i)2-s + (0.900 + 0.433i)3-s + (1.04 − 0.503i)4-s + (−0.0634 − 0.277i)5-s + (−0.203 + 0.892i)6-s + (−4.54 − 2.18i)7-s + (1.80 + 2.26i)8-s + (0.623 + 0.781i)9-s + (0.235 − 0.113i)10-s + (−2.60 + 3.26i)11-s + 1.16·12-s + (1.41 − 1.77i)13-s + (1.02 − 4.49i)14-s + (0.0634 − 0.277i)15-s + (−0.204 + 0.256i)16-s − 3.40·17-s + ⋯
L(s)  = 1  + (0.144 + 0.631i)2-s + (0.520 + 0.250i)3-s + (0.523 − 0.251i)4-s + (−0.0283 − 0.124i)5-s + (−0.0831 + 0.364i)6-s + (−1.71 − 0.826i)7-s + (0.638 + 0.800i)8-s + (0.207 + 0.260i)9-s + (0.0743 − 0.0358i)10-s + (−0.786 + 0.985i)11-s + 0.335·12-s + (0.393 − 0.493i)13-s + (0.274 − 1.20i)14-s + (0.0163 − 0.0717i)15-s + (−0.0510 + 0.0640i)16-s − 0.826·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.773 - 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12320 + 0.400903i\)
\(L(\frac12)\) \(\approx\) \(1.12320 + 0.400903i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 4.81i)T \)
good2 \( 1 + (-0.203 - 0.892i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.0634 + 0.277i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (4.54 + 2.18i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (2.60 - 3.26i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.41 + 1.77i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 3.40T + 17T^{2} \)
19 \( 1 + (0.768 - 0.369i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-1.61 + 7.06i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.728 - 3.18i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (-0.237 - 0.297i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 5.56T + 41T^{2} \)
43 \( 1 + (1.75 - 7.70i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-4.07 + 5.10i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.643 + 2.82i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 4.68T + 59T^{2} \)
61 \( 1 + (-6.25 - 3.01i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (1.77 + 2.22i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (7.16 - 8.98i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-2.51 + 11.0i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (4.22 + 5.29i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (5.65 - 2.72i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.26 + 5.56i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (0.453 - 0.218i)T + (60.4 - 75.8i)T^{2} \)
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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

−14.49547890865336996274585757184, −13.29710577721514152244328050244, −12.60117652900676482200157105995, −10.60024006133918225112399872517, −10.15242427152875456195570961639, −8.615049088521025700493071100412, −7.17835237114699243479607450892, −6.45914288867065132690537763409, −4.69151722685619570225598728961, −2.86944808053454088208169353824, 2.59294275511740168070333641521, 3.52307780115957818715897757023, 6.04617873448739441364004847165, 7.08879819042059365802511269525, 8.648507334275361163928768731747, 9.720875159998078997906835981996, 10.97287172402162558728539641906, 12.02825511134601057031413111394, 13.13164540119560641575419823775, 13.49940833364045067197241688194

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