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Properties

Label 2-87-29.7-c1-0-4
Degree $2$
Conductor $87$
Sign $0.0372 + 0.999i$
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  + (−0.399 − 1.74i)2-s + (0.900 + 0.433i)3-s + (−1.10 + 0.529i)4-s + (−0.299 − 1.31i)5-s + (0.399 − 1.74i)6-s + (1.00 + 0.483i)7-s + (−0.871 − 1.09i)8-s + (0.623 + 0.781i)9-s + (−2.17 + 1.04i)10-s + (−1.53 + 1.93i)11-s − 1.22·12-s + (−1.75 + 2.20i)13-s + (0.444 − 1.94i)14-s + (0.299 − 1.31i)15-s + (−3.08 + 3.87i)16-s + 5.50·17-s + ⋯
L(s)  = 1  + (−0.282 − 1.23i)2-s + (0.520 + 0.250i)3-s + (−0.550 + 0.264i)4-s + (−0.133 − 0.586i)5-s + (0.163 − 0.714i)6-s + (0.379 + 0.182i)7-s + (−0.308 − 0.386i)8-s + (0.207 + 0.260i)9-s + (−0.687 + 0.331i)10-s + (−0.464 + 0.582i)11-s − 0.352·12-s + (−0.487 + 0.610i)13-s + (0.118 − 0.520i)14-s + (0.0772 − 0.338i)15-s + (−0.771 + 0.967i)16-s + 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0372 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0372 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.0372 + 0.999i$
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: \((2,\ 87,\ (\ :1/2),\ 0.0372 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709558 - 0.683571i\)
\(L(\frac12)\) \(\approx\) \(0.709558 - 0.683571i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.29 + 0.981i)T \)
good2 \( 1 + (0.399 + 1.74i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.299 + 1.31i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-1.00 - 0.483i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.53 - 1.93i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.75 - 2.20i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 - 5.50T + 17T^{2} \)
19 \( 1 + (-0.318 + 0.153i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.18 - 5.18i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.990 - 4.33i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (6.00 + 7.52i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + 9.05T + 41T^{2} \)
43 \( 1 + (-2.01 + 8.82i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-2.24 + 2.81i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.46 - 6.41i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 6.26T + 59T^{2} \)
61 \( 1 + (0.106 + 0.0512i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-8.91 - 11.1i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-4.68 + 5.86i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-3.54 + 15.5i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-0.160 - 0.200i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-5.52 + 2.65i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.66 + 7.31i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (6.37 - 3.07i)T + (60.4 - 75.8i)T^{2} \)
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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

−13.74349436034626932759294383639, −12.47299811079381710247712652747, −11.91049142222315017987804321054, −10.57970530246585255993190735620, −9.678802075328873372790333170407, −8.749281084830674980893654728294, −7.37124917646129727597585952061, −5.15520813837550531855458318983, −3.59486991487819349566625928273, −1.91113329301265944928779355277, 3.02280751128944763674251069583, 5.28678870161059551461664721660, 6.63102492328295062507112919443, 7.73673413998218321588103746075, 8.314037887750172186842692903276, 9.833408095384208736425607964526, 11.14166482605076096030060578330, 12.49862358696200197818924384935, 13.89833404375414612127693639093, 14.66609146321388820445480692163

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