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Properties

Label 2-87-29.20-c1-0-2
Degree $2$
Conductor $87$
Sign $0.954 - 0.299i$
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  + (−1.04 + 0.500i)2-s + (−0.623 − 0.781i)3-s + (−0.416 + 0.521i)4-s + (3.10 − 1.49i)5-s + (1.04 + 0.500i)6-s + (2.81 + 3.53i)7-s + (0.685 − 3.00i)8-s + (−0.222 + 0.974i)9-s + (−2.48 + 3.11i)10-s + (0.123 + 0.540i)11-s + 0.667·12-s + (−0.614 − 2.69i)13-s + (−4.69 − 2.26i)14-s + (−3.10 − 1.49i)15-s + (0.493 + 2.16i)16-s − 2.99·17-s + ⋯
L(s)  = 1  + (−0.735 + 0.354i)2-s + (−0.359 − 0.451i)3-s + (−0.208 + 0.260i)4-s + (1.38 − 0.669i)5-s + (0.424 + 0.204i)6-s + (1.06 + 1.33i)7-s + (0.242 − 1.06i)8-s + (−0.0741 + 0.324i)9-s + (−0.785 + 0.984i)10-s + (0.0371 + 0.162i)11-s + 0.192·12-s + (−0.170 − 0.746i)13-s + (−1.25 − 0.604i)14-s + (−0.802 − 0.386i)15-s + (0.123 + 0.541i)16-s − 0.726·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.954 - 0.299i$
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: \((2,\ 87,\ (\ :1/2),\ 0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729646 + 0.111718i\)
\(L(\frac12)\) \(\approx\) \(0.729646 + 0.111718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.37 + 0.265i)T \)
good2 \( 1 + (1.04 - 0.500i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (-3.10 + 1.49i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-2.81 - 3.53i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.123 - 0.540i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.614 + 2.69i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 2.99T + 17T^{2} \)
19 \( 1 + (0.173 - 0.217i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (0.836 + 0.402i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (8.21 - 3.95i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-1.52 + 6.70i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 2.85T + 41T^{2} \)
43 \( 1 + (-10.1 - 4.86i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (0.933 + 4.09i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.87 + 0.901i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + (4.50 + 5.65i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-2.73 + 11.9i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.530 - 2.32i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.822 - 0.396i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (0.460 - 2.01i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (6.83 - 8.57i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-6.13 + 2.95i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (2.19 - 2.74i)T + (-21.5 - 94.5i)T^{2} \)
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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

−14.17912554245879296819183477160, −12.93836797014185057406395240496, −12.42730590728811132991081768202, −10.89611068764399141558334075871, −9.402907339106312293078091655613, −8.791730475458109670612653961113, −7.65538138934954537305403392104, −5.99677699763847541971715031521, −5.03357936346255450219982332670, −1.93482905611807858650630086516, 1.80351932704876151773841839122, 4.48162903083060553061157483701, 5.83750007089789818103747118501, 7.33727245897799606280269382787, 9.012293052843298308445987640552, 9.913861828082166033268532562154, 10.78204111084409700682967660388, 11.27008733233902995908185165608, 13.46572944405036329529590712506, 14.12796569723191849447857700651

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