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Properties

Label 2-87-29.23-c1-0-5
Degree $2$
Conductor $87$
Sign $0.238 + 0.971i$
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  + (1.61 − 2.02i)2-s + (−0.222 + 0.974i)3-s + (−1.04 − 4.58i)4-s + (−0.716 + 0.898i)5-s + (1.61 + 2.02i)6-s + (−0.615 + 2.69i)7-s + (−6.29 − 3.03i)8-s + (−0.900 − 0.433i)9-s + (0.661 + 2.89i)10-s + (3.92 − 1.88i)11-s + 4.70·12-s + (−5.07 + 2.44i)13-s + (4.46 + 5.59i)14-s + (−0.716 − 0.898i)15-s + (−7.83 + 3.77i)16-s + 2.41·17-s + ⋯
L(s)  = 1  + (1.14 − 1.43i)2-s + (−0.128 + 0.562i)3-s + (−0.523 − 2.29i)4-s + (−0.320 + 0.401i)5-s + (0.658 + 0.826i)6-s + (−0.232 + 1.01i)7-s + (−2.22 − 1.07i)8-s + (−0.300 − 0.144i)9-s + (0.209 + 0.916i)10-s + (1.18 − 0.569i)11-s + 1.35·12-s + (−1.40 + 0.677i)13-s + (1.19 + 1.49i)14-s + (−0.184 − 0.231i)15-s + (−1.95 + 0.943i)16-s + 0.585·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.238 + 0.971i$
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: \((2,\ 87,\ (\ :1/2),\ 0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12695 - 0.883566i\)
\(L(\frac12)\) \(\approx\) \(1.12695 - 0.883566i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.53 - 2.90i)T \)
good2 \( 1 + (-1.61 + 2.02i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (0.716 - 0.898i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.615 - 2.69i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-3.92 + 1.88i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (5.07 - 2.44i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 2.41T + 17T^{2} \)
19 \( 1 + (0.416 + 1.82i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (5.49 + 6.89i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-0.972 + 1.22i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (6.99 + 3.36i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 - 3.16T + 41T^{2} \)
43 \( 1 + (0.912 + 1.14i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.321 - 0.155i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.01 - 1.27i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 4.85T + 59T^{2} \)
61 \( 1 + (0.346 - 1.51i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-8.71 - 4.19i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-7.37 + 3.54i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-7.09 - 8.89i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-7.32 - 3.52i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-0.107 - 0.469i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (1.24 - 1.56i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (1.03 + 4.51i)T + (-87.3 + 42.0i)T^{2} \)
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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

−14.17287562518573824608123885542, −12.37568178351223711068871158640, −12.02828332861137323908720052767, −11.05149480962974125964177867677, −9.928918197062244174519750232199, −8.986276119359527707921251015141, −6.44304510033463415173484092572, −5.12446628284446948598835557297, −3.87408855994787623833612340101, −2.54866113577040171331492330518, 3.75731764435968774470892816384, 4.95204838226911371117602036292, 6.36847925704704372442647373315, 7.35232893365502883481237228043, 8.071843040739414971117056122167, 9.868619263792397491667947412371, 12.08286885366379150780480349447, 12.38603921139558699823896182455, 13.71309281537880333588503357547, 14.27616951443288338087874889876

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