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Properties

Label 2-87-29.20-c1-0-3
Degree $2$
Conductor $87$
Sign $0.980 - 0.198i$
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  + (0.626 − 0.301i)2-s + (0.623 + 0.781i)3-s + (−0.945 + 1.18i)4-s + (1.81 − 0.875i)5-s + (0.626 + 0.301i)6-s + (−1.49 − 1.87i)7-s + (−0.543 + 2.38i)8-s + (−0.222 + 0.974i)9-s + (0.874 − 1.09i)10-s + (0.213 + 0.933i)11-s − 1.51·12-s + (−1.45 − 6.36i)13-s + (−1.49 − 0.722i)14-s + (1.81 + 0.875i)15-s + (−0.297 − 1.30i)16-s − 3.81·17-s + ⋯
L(s)  = 1  + (0.442 − 0.213i)2-s + (0.359 + 0.451i)3-s + (−0.472 + 0.593i)4-s + (0.813 − 0.391i)5-s + (0.255 + 0.123i)6-s + (−0.564 − 0.707i)7-s + (−0.192 + 0.842i)8-s + (−0.0741 + 0.324i)9-s + (0.276 − 0.346i)10-s + (0.0642 + 0.281i)11-s − 0.437·12-s + (−0.403 − 1.76i)13-s + (−0.400 − 0.192i)14-s + (0.469 + 0.226i)15-s + (−0.0742 − 0.325i)16-s − 0.925·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.980 - 0.198i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21946 + 0.122332i\)
\(L(\frac12)\) \(\approx\) \(1.21946 + 0.122332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (-5.32 - 0.822i)T \)
good2 \( 1 + (-0.626 + 0.301i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-1.81 + 0.875i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (1.49 + 1.87i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.213 - 0.933i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (1.45 + 6.36i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 + (2.69 - 3.37i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-4.85 - 2.33i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (2.46 - 1.18i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-0.414 + 1.81i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + (-3.33 - 1.60i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.719 - 3.15i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (2.04 - 0.983i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 9.30T + 59T^{2} \)
61 \( 1 + (-1.95 - 2.45i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (2.69 - 11.7i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-1.02 - 4.48i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (8.19 + 3.94i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (0.954 - 4.18i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (-10.0 + 12.6i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (13.9 - 6.70i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-9.82 + 12.3i)T + (-21.5 - 94.5i)T^{2} \)
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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

−14.00686756212010650108514069515, −13.04551241911722362111960096692, −12.66241706770323601678831358789, −10.84394895185136316765051470202, −9.821714634851823745537148712805, −8.802768864765464308260464492586, −7.53980768272226008651777164328, −5.63910136569424856062767399787, −4.35143892212928684892717348392, −2.93429556732389874933229058579, 2.38999204025373528884186425916, 4.52827333859278540616598365948, 6.17476363171916735581843429900, 6.74480056948357067830448435816, 8.990600444375428211614297167666, 9.403874665916473964606072038548, 10.89192969477129782820064787031, 12.38334003953773202386472265489, 13.39249347464210750220078345830, 14.06819514434524724865526330611

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