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Properties

Label 2-87-29.20-c1-0-4
Degree $2$
Conductor $87$
Sign $-0.422 + 0.906i$
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  + (−1.86 + 0.899i)2-s + (−0.623 − 0.781i)3-s + (1.43 − 1.79i)4-s + (−1.70 + 0.822i)5-s + (1.86 + 0.899i)6-s + (−2.93 − 3.68i)7-s + (−0.137 + 0.604i)8-s + (−0.222 + 0.974i)9-s + (2.45 − 3.07i)10-s + (−0.832 − 3.64i)11-s − 2.29·12-s + (0.250 + 1.09i)13-s + (8.80 + 4.23i)14-s + (1.70 + 0.822i)15-s + (0.737 + 3.22i)16-s − 5.60·17-s + ⋯
L(s)  = 1  + (−1.32 + 0.636i)2-s + (−0.359 − 0.451i)3-s + (0.716 − 0.898i)4-s + (−0.763 + 0.367i)5-s + (0.762 + 0.367i)6-s + (−1.11 − 1.39i)7-s + (−0.0487 + 0.213i)8-s + (−0.0741 + 0.324i)9-s + (0.774 − 0.971i)10-s + (−0.250 − 1.09i)11-s − 0.663·12-s + (0.0694 + 0.304i)13-s + (2.35 + 1.13i)14-s + (0.440 + 0.212i)15-s + (0.184 + 0.807i)16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.422 + 0.906i$
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: \((2,\ 87,\ (\ :1/2),\ -0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0960117 - 0.150742i\)
\(L(\frac12)\) \(\approx\) \(0.0960117 - 0.150742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.82 + 3.79i)T \)
good2 \( 1 + (1.86 - 0.899i)T + (1.24 - 1.56i)T^{2} \)
5 \( 1 + (1.70 - 0.822i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (2.93 + 3.68i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (0.832 + 3.64i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.250 - 1.09i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 5.60T + 17T^{2} \)
19 \( 1 + (1.20 - 1.51i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-3.89 - 1.87i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (0.874 - 0.421i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-2.21 + 9.71i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 - 1.59T + 41T^{2} \)
43 \( 1 + (4.08 + 1.96i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.30 + 5.71i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (4.64 - 2.23i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 5.71T + 59T^{2} \)
61 \( 1 + (6.51 + 8.16i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-0.883 + 3.86i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.0665 - 0.291i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-13.6 - 6.55i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-3.07 + 13.4i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (-1.81 + 2.28i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (6.56 - 3.16i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (7.99 - 10.0i)T + (-21.5 - 94.5i)T^{2} \)
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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

−13.76510114089098072099725010068, −12.93638418520070515313129313397, −11.19694399901726266972220934252, −10.57431105887403485800104486571, −9.277870869383365607802339168102, −8.010582384727180109768103075881, −7.07236362740617034961699628309, −6.35899788416307813139331277134, −3.77783770503437198700932372242, −0.33663499338349850951616216263, 2.69930477157737553204957959417, 4.84097997171007912815234348814, 6.66394405070562912935527997123, 8.337579529425072061015132709781, 9.151671938806780205743081172052, 9.997920265924044007766611695230, 11.17089615573558642086859872206, 12.12620799498255366611452399804, 12.86705748010902315276593109137, 15.14261221794553540389408834197

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