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Properties

Label 2-87-29.23-c1-0-3
Degree $2$
Conductor $87$
Sign $0.994 + 0.103i$
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  + (−0.0518 + 0.0650i)2-s + (0.222 − 0.974i)3-s + (0.443 + 1.94i)4-s + (1.20 − 1.50i)5-s + (0.0518 + 0.0650i)6-s + (−0.0765 + 0.335i)7-s + (−0.299 − 0.144i)8-s + (−0.900 − 0.433i)9-s + (0.0357 + 0.156i)10-s + (2.18 − 1.05i)11-s + 1.99·12-s + (−3.27 + 1.57i)13-s + (−0.0178 − 0.0223i)14-s + (−1.20 − 1.50i)15-s + (−3.56 + 1.71i)16-s − 6.15·17-s + ⋯
L(s)  = 1  + (−0.0366 + 0.0459i)2-s + (0.128 − 0.562i)3-s + (0.221 + 0.971i)4-s + (0.538 − 0.674i)5-s + (0.0211 + 0.0265i)6-s + (−0.0289 + 0.126i)7-s + (−0.105 − 0.0509i)8-s + (−0.300 − 0.144i)9-s + (0.0112 + 0.0494i)10-s + (0.658 − 0.317i)11-s + 0.575·12-s + (−0.907 + 0.436i)13-s + (−0.00476 − 0.00597i)14-s + (−0.310 − 0.389i)15-s + (−0.891 + 0.429i)16-s − 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.994 + 0.103i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05541 - 0.0545208i\)
\(L(\frac12)\) \(\approx\) \(1.05541 - 0.0545208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (4.26 - 3.29i)T \)
good2 \( 1 + (0.0518 - 0.0650i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (-1.20 + 1.50i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.0765 - 0.335i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-2.18 + 1.05i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.27 - 1.57i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 6.15T + 17T^{2} \)
19 \( 1 + (-0.300 - 1.31i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (2.16 + 2.70i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-6.85 + 8.59i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (-2.78 - 1.33i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 - 5.28T + 41T^{2} \)
43 \( 1 + (0.00526 + 0.00659i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-5.22 + 2.51i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (2.58 - 3.23i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 5.76T + 59T^{2} \)
61 \( 1 + (-1.38 + 6.08i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (9.92 + 4.77i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (11.5 - 5.57i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-0.286 - 0.359i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-11.2 - 5.41i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (0.640 + 2.80i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (7.62 - 9.55i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-0.606 - 2.65i)T + (-87.3 + 42.0i)T^{2} \)
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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

−13.88625020839372426783470706555, −13.04612654294961150131848483169, −12.20951197224154527463370358444, −11.28268786247941478890160201829, −9.408569874164116857878163937355, −8.605981583085119117281117588126, −7.35118704892474149938195512250, −6.18236823293577027874684331002, −4.31830425762404274827130986333, −2.32574764180977286338758692199, 2.42845099258208896765894669285, 4.56719057115337065583188079935, 6.00361210190932933877974384582, 7.09332890571311192474067950348, 9.041094055793296856686353436235, 9.975440516733443496364393678535, 10.69629314685900320849090979998, 11.80263783270728393119184326482, 13.49621310297348168381822310243, 14.39736827457890762328597777300

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