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Properties

Label 2-87-29.25-c1-0-0
Degree $2$
Conductor $87$
Sign $-0.990 + 0.138i$
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  + (−0.563 + 2.47i)2-s + (−0.900 + 0.433i)3-s + (−3.98 − 1.91i)4-s + (−0.242 + 1.06i)5-s + (−0.563 − 2.47i)6-s + (−1.55 + 0.750i)7-s + (3.82 − 4.79i)8-s + (0.623 − 0.781i)9-s + (−2.48 − 1.19i)10-s + (3.92 + 4.92i)11-s + 4.42·12-s + (−0.797 − 1.00i)13-s + (−0.975 − 4.27i)14-s + (−0.242 − 1.06i)15-s + (4.18 + 5.24i)16-s − 5.08·17-s + ⋯
L(s)  = 1  + (−0.398 + 1.74i)2-s + (−0.520 + 0.250i)3-s + (−1.99 − 0.959i)4-s + (−0.108 + 0.475i)5-s + (−0.230 − 1.00i)6-s + (−0.589 + 0.283i)7-s + (1.35 − 1.69i)8-s + (0.207 − 0.260i)9-s + (−0.786 − 0.378i)10-s + (1.18 + 1.48i)11-s + 1.27·12-s + (−0.221 − 0.277i)13-s + (−0.260 − 1.14i)14-s + (−0.0625 − 0.274i)15-s + (1.04 + 1.31i)16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.990 + 0.138i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ -0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0392087 - 0.564053i\)
\(L(\frac12)\) \(\approx\) \(0.0392087 - 0.564053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (1.89 + 5.04i)T \)
good2 \( 1 + (0.563 - 2.47i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (0.242 - 1.06i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (1.55 - 0.750i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-3.92 - 4.92i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.797 + 1.00i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 5.08T + 17T^{2} \)
19 \( 1 + (-5.88 - 2.83i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.00 - 4.41i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-1.52 + 6.68i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-3.84 + 4.82i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 + (0.407 + 1.78i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-0.945 - 1.18i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.839 + 3.67i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 8.72T + 59T^{2} \)
61 \( 1 + (-2.78 + 1.33i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (1.49 - 1.87i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-5.82 - 7.30i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.400 - 1.75i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (1.78 - 2.24i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (5.94 + 2.86i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.744 - 3.26i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (2.04 + 0.983i)T + (60.4 + 75.8i)T^{2} \)
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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

−15.16118050022843159921866083262, −14.19906347700537839649856219169, −12.87310152589461963112684182517, −11.50482433126427757286091545137, −9.742431912517094940363077985279, −9.324624227901577572686792548341, −7.55885984262009143855116758703, −6.77466013218414382650062451083, −5.73919089011354904261719852341, −4.30223817013189553533458349916, 0.938251382334138438353262608340, 3.17500430035784360813652181235, 4.64437198777814347121229335412, 6.63001479973554901091548056054, 8.648553435709736852854558542037, 9.325531234427143624171984320365, 10.70883349119752888891978463045, 11.43112893218702336535556728576, 12.28698186215596000945584962041, 13.23732712008420936976416394457

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