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Properties

Label 2-87-87.17-c1-0-3
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: $\chi_{87} (17, \cdot )$
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

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

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