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Properties

Label 2-87-87.41-c1-0-4
Degree $2$
Conductor $87$
Sign $0.944 - 0.327i$
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  + (−0.366 + 0.366i)2-s + (1.5 − 0.866i)3-s + 1.73i·4-s − 0.267·5-s + (−0.232 + 0.866i)6-s + 0.732·7-s + (−1.36 − 1.36i)8-s + (1.5 − 2.59i)9-s + (0.0980 − 0.0980i)10-s + (−1.36 + 1.36i)11-s + (1.49 + 2.59i)12-s − 2.26i·13-s + (−0.267 + 0.267i)14-s + (−0.401 + 0.232i)15-s − 2.46·16-s + (−2.73 + 2.73i)17-s + ⋯
L(s)  = 1  + (−0.258 + 0.258i)2-s + (0.866 − 0.499i)3-s + 0.866i·4-s − 0.119·5-s + (−0.0947 + 0.353i)6-s + 0.276·7-s + (−0.482 − 0.482i)8-s + (0.5 − 0.866i)9-s + (0.0310 − 0.0310i)10-s + (−0.411 + 0.411i)11-s + (0.433 + 0.749i)12-s − 0.629i·13-s + (−0.0716 + 0.0716i)14-s + (−0.103 + 0.0599i)15-s − 0.616·16-s + (−0.662 + 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03710 + 0.174475i\)
\(L(\frac12)\) \(\approx\) \(1.03710 + 0.174475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
29 \( 1 + (2 - 5i)T \)
good2 \( 1 + (0.366 - 0.366i)T - 2iT^{2} \)
5 \( 1 + 0.267T + 5T^{2} \)
7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 + (1.36 - 1.36i)T - 11iT^{2} \)
13 \( 1 + 2.26iT - 13T^{2} \)
17 \( 1 + (2.73 - 2.73i)T - 17iT^{2} \)
19 \( 1 + (1.73 + 1.73i)T + 19iT^{2} \)
23 \( 1 + 5.46iT - 23T^{2} \)
31 \( 1 + (-2.36 - 2.36i)T + 31iT^{2} \)
37 \( 1 + (-3.46 + 3.46i)T - 37iT^{2} \)
41 \( 1 + (-5.19 - 5.19i)T + 41iT^{2} \)
43 \( 1 + (-8.56 - 8.56i)T + 43iT^{2} \)
47 \( 1 + (-7.83 - 7.83i)T + 47iT^{2} \)
53 \( 1 + 9.92iT - 53T^{2} \)
59 \( 1 + 7.46iT - 59T^{2} \)
61 \( 1 + (2 + 2i)T + 61iT^{2} \)
67 \( 1 - 4.92iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + (4 - 4i)T - 73iT^{2} \)
79 \( 1 + (6.83 + 6.83i)T + 79iT^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (5 - 5i)T - 89iT^{2} \)
97 \( 1 + (4.26 - 4.26i)T - 97iT^{2} \)
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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

−14.30106767555518501831863607997, −12.88001165732150382086990163099, −12.64624079342826686012076323816, −11.06347921604589120247420001415, −9.492457661852572178851926293300, −8.372966979678730585969204056346, −7.71261809984032131594904448391, −6.51465495053721839068258632211, −4.20940093610889004302015077257, −2.62553781196772683092613592498, 2.25748004816738255987187310989, 4.23761528155496304471603523463, 5.72610350175368367215024739514, 7.54754222236557277988611637100, 8.851157268663388350528539366246, 9.669609052780942083843827391482, 10.73126848803377284096577615456, 11.67997156820131818952387599408, 13.53729277834300241770059967980, 14.06069532665788804976991301927

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