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Properties

Label 2-87-29.16-c1-0-1
Degree $2$
Conductor $87$
Sign $0.526 + 0.850i$
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  + (−2.50 − 1.20i)2-s + (0.623 − 0.781i)3-s + (3.56 + 4.46i)4-s + (2.28 + 1.10i)5-s + (−2.50 + 1.20i)6-s + (0.527 − 0.661i)7-s + (−2.29 − 10.0i)8-s + (−0.222 − 0.974i)9-s + (−4.39 − 5.51i)10-s + (−0.279 + 1.22i)11-s + 5.71·12-s + (0.494 − 2.16i)13-s + (−2.11 + 1.02i)14-s + (2.28 − 1.10i)15-s + (−3.83 + 16.8i)16-s + 3.75·17-s + ⋯
L(s)  = 1  + (−1.76 − 0.852i)2-s + (0.359 − 0.451i)3-s + (1.78 + 2.23i)4-s + (1.02 + 0.492i)5-s + (−1.02 + 0.492i)6-s + (0.199 − 0.250i)7-s + (−0.812 − 3.55i)8-s + (−0.0741 − 0.324i)9-s + (−1.38 − 1.74i)10-s + (−0.0843 + 0.369i)11-s + 1.65·12-s + (0.137 − 0.600i)13-s + (−0.566 + 0.272i)14-s + (0.590 − 0.284i)15-s + (−0.959 + 4.20i)16-s + 0.910·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.526 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522023 - 0.290891i\)
\(L(\frac12)\) \(\approx\) \(0.522023 - 0.290891i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.24 - 3.31i)T \)
good2 \( 1 + (2.50 + 1.20i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (-2.28 - 1.10i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-0.527 + 0.661i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (0.279 - 1.22i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.494 + 2.16i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 3.75T + 17T^{2} \)
19 \( 1 + (1.84 + 2.31i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (5.78 - 2.78i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (0.518 + 0.249i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-0.893 - 3.91i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + 9.29T + 41T^{2} \)
43 \( 1 + (8.60 - 4.14i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.626 + 2.74i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (3.55 + 1.71i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 2.24T + 59T^{2} \)
61 \( 1 + (2.97 - 3.72i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-0.0320 - 0.140i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.13 + 4.99i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (1.24 - 0.598i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (0.0599 + 0.262i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (4.26 + 5.34i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-4.15 - 2.00i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (2.88 + 3.61i)T + (-21.5 + 94.5i)T^{2} \)
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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

−13.77059105687630994354561604096, −12.65472005504019981324689488663, −11.60270799855220078889313773971, −10.29866760983284353268012804356, −9.875936978016397808563769683782, −8.558402936610543254441404295949, −7.59323603489120602776404489095, −6.43143307308723813462233884982, −3.09154966484573071098308588733, −1.71581533436500632589262428423, 1.91455088683630914974446783842, 5.39452862743073778796032762475, 6.40624947595454129014980299109, 8.033456228901254714643979793390, 8.778360091275642906152686775005, 9.782352911471439748439430531177, 10.38156616128725646636368702199, 11.81781348713223524384667220142, 13.86074442570908331572554390460, 14.63677874606697569757893119269

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