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Properties

Label 2-87-87.86-c4-0-15
Degree $2$
Conductor $87$
Sign $0.811 + 0.584i$
Analytic cond. $8.99318$
Root an. cond. $2.99886$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.29·2-s + (−1.45 + 8.88i)3-s − 5.16·4-s − 6.84i·5-s + (4.77 − 29.2i)6-s − 58.5·7-s + 69.6·8-s + (−76.7 − 25.7i)9-s + 22.5i·10-s − 62.4·11-s + (7.49 − 45.8i)12-s + 135.·13-s + 192.·14-s + (60.7 + 9.93i)15-s − 146.·16-s + 362.·17-s + ⋯
L(s)  = 1  − 0.822·2-s + (−0.161 + 0.986i)3-s − 0.322·4-s − 0.273i·5-s + (0.132 − 0.812i)6-s − 1.19·7-s + 1.08·8-s + (−0.947 − 0.318i)9-s + 0.225i·10-s − 0.516·11-s + (0.0520 − 0.318i)12-s + 0.801·13-s + 0.982·14-s + (0.270 + 0.0441i)15-s − 0.572·16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(8.99318\)
Root analytic conductor: \(2.99886\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :2),\ 0.811 + 0.584i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.523159 - 0.168792i\)
\(L(\frac12)\) \(\approx\) \(0.523159 - 0.168792i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.45 - 8.88i)T \)
29 \( 1 + (-375. + 752. i)T \)
good2 \( 1 + 3.29T + 16T^{2} \)
5 \( 1 + 6.84iT - 625T^{2} \)
7 \( 1 + 58.5T + 2.40e3T^{2} \)
11 \( 1 + 62.4T + 1.46e4T^{2} \)
13 \( 1 - 135.T + 2.85e4T^{2} \)
17 \( 1 - 362.T + 8.35e4T^{2} \)
19 \( 1 - 168. iT - 1.30e5T^{2} \)
23 \( 1 + 889. iT - 2.79e5T^{2} \)
31 \( 1 - 156. iT - 9.23e5T^{2} \)
37 \( 1 - 1.49e3iT - 1.87e6T^{2} \)
41 \( 1 + 90.8T + 2.82e6T^{2} \)
43 \( 1 + 2.74e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.05e3T + 4.87e6T^{2} \)
53 \( 1 - 2.79e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.40e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.21e3iT - 1.38e7T^{2} \)
67 \( 1 + 2.42e3T + 2.01e7T^{2} \)
71 \( 1 + 1.59e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.00e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.22e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.06e4iT - 4.74e7T^{2} \)
89 \( 1 - 5.03e3T + 6.27e7T^{2} \)
97 \( 1 - 1.32e4iT - 8.85e7T^{2} \)
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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

−13.34866471527712978881628733151, −12.22765909953636960855991303193, −10.55577125691262265561436789639, −10.06837915528370292278365381698, −9.033022431415146397930132216361, −8.134957391173179345721249940555, −6.25639720242140882865249073310, −4.79882390539986701423535047128, −3.35519907569250411305164865587, −0.45808557825477366411876385188, 1.09198999562597453001726803870, 3.21077908112246488198380872404, 5.53286303032459738751364515602, 6.89513566938234314217038398783, 7.889113876665576454651824109586, 9.059583972071300325946190910208, 10.13413315117276013909894786551, 11.27359360966227246939860461132, 12.74921520594209395886504823748, 13.31419606539163895476487850651

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