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Properties

Label 2-87-29.7-c1-0-3
Degree $2$
Conductor $87$
Sign $0.894 + 0.446i$
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  + (0.0321 + 0.140i)2-s + (−0.900 − 0.433i)3-s + (1.78 − 0.858i)4-s + (−0.345 − 1.51i)5-s + (0.0321 − 0.140i)6-s + (1.37 + 0.662i)7-s + (0.358 + 0.449i)8-s + (0.623 + 0.781i)9-s + (0.202 − 0.0973i)10-s + (−0.478 + 0.600i)11-s − 1.97·12-s + (−1.63 + 2.04i)13-s + (−0.0490 + 0.215i)14-s + (−0.345 + 1.51i)15-s + (2.41 − 3.02i)16-s − 3.51·17-s + ⋯
L(s)  = 1  + (0.0227 + 0.0995i)2-s + (−0.520 − 0.250i)3-s + (0.891 − 0.429i)4-s + (−0.154 − 0.677i)5-s + (0.0131 − 0.0574i)6-s + (0.520 + 0.250i)7-s + (0.126 + 0.158i)8-s + (0.207 + 0.260i)9-s + (0.0639 − 0.0307i)10-s + (−0.144 + 0.181i)11-s − 0.571·12-s + (−0.452 + 0.567i)13-s + (−0.0131 + 0.0574i)14-s + (−0.0892 + 0.391i)15-s + (0.604 − 0.757i)16-s − 0.852·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ 0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966207 - 0.227907i\)
\(L(\frac12)\) \(\approx\) \(0.966207 - 0.227907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.40 + 4.17i)T \)
good2 \( 1 + (-0.0321 - 0.140i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (0.345 + 1.51i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-1.37 - 0.662i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (0.478 - 0.600i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.63 - 2.04i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 3.51T + 17T^{2} \)
19 \( 1 + (4.72 - 2.27i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.89 - 8.28i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.315 - 1.38i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + (1.87 + 2.35i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 5.79T + 41T^{2} \)
43 \( 1 + (0.955 - 4.18i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-1.10 + 1.38i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.50 + 6.60i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 + (12.3 + 5.95i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (4.54 + 5.70i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (2.99 - 3.76i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.04 - 8.96i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-0.340 - 0.426i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (10.4 - 5.05i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (3.06 + 13.4i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-1.60 + 0.770i)T + (60.4 - 75.8i)T^{2} \)
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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

−14.22963467935774868816824210424, −12.84320591958800579238552009190, −11.85508620145153438733724530841, −11.13295682263897489253077756894, −9.874168875943196710144665590460, −8.356993459956776412929883665482, −7.10558541090672038179450544680, −5.89706204145986040926853953899, −4.64887843044907221215178461780, −1.92334256256028243820663637251, 2.67901717903275616668530623241, 4.47709277902001567762267936328, 6.30110974956992593324082866786, 7.22975190344445606786796815804, 8.544864067579210678659287418422, 10.62252392553050094239588145976, 10.75911588678619403186727746007, 12.00113659042793307969331366146, 12.95989344611590501622438245324, 14.55375078715726297853212233109

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