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Properties

Label 2-867-51.41-c1-0-39
Degree $2$
Conductor $867$
Sign $-0.0604 + 0.998i$
Analytic cond. $6.92302$
Root an. cond. $2.63116$
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  + (−0.962 + 1.44i)3-s + (−1.41 − 1.41i)4-s + (−0.651 + 3.27i)7-s + (−1.14 − 2.77i)9-s + (3.39 − 0.675i)12-s + (−2.06 + 2.06i)13-s + 4.00i·16-s + (3.33 − 8.03i)19-s + (−4.08 − 4.08i)21-s + (−4.61 + 1.91i)25-s + (5.09 + 1.01i)27-s + (5.55 − 3.70i)28-s + (−6.98 − 4.66i)31-s + (−2.29 + 5.54i)36-s + (6.51 − 9.75i)37-s + ⋯
L(s)  = 1  + (−0.555 + 0.831i)3-s + (−0.707 − 0.707i)4-s + (−0.246 + 1.23i)7-s + (−0.382 − 0.923i)9-s + (0.980 − 0.195i)12-s + (−0.572 + 0.572i)13-s + 1.00i·16-s + (0.764 − 1.84i)19-s + (−0.892 − 0.892i)21-s + (−0.923 + 0.382i)25-s + (0.980 + 0.195i)27-s + (1.04 − 0.700i)28-s + (−1.25 − 0.837i)31-s + (−0.382 + 0.923i)36-s + (1.07 − 1.60i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0604 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0604 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.0604 + 0.998i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.0604 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267219 - 0.283891i\)
\(L(\frac12)\) \(\approx\) \(0.267219 - 0.283891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.962 - 1.44i)T \)
17 \( 1 \)
good2 \( 1 + (1.41 + 1.41i)T^{2} \)
5 \( 1 + (4.61 - 1.91i)T^{2} \)
7 \( 1 + (0.651 - 3.27i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (2.06 - 2.06i)T - 13iT^{2} \)
19 \( 1 + (-3.33 + 8.03i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (6.98 + 4.66i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (-6.51 + 9.75i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (4.44 + 10.7i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.68 + 0.335i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + 14.4iT - 67T^{2} \)
71 \( 1 + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (0.337 + 1.69i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (4.32 - 2.88i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (1.58 - 0.315i)T + (89.6 - 37.1i)T^{2} \)
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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

−9.563497256199080077827657449306, −9.402019570564921404344278875947, −8.753806522938931337493607294658, −7.25594194559810181962766069451, −6.09353119026669320483212288746, −5.44651896087789782675502881462, −4.79259248266101759527271497195, −3.72963019760262170167836358904, −2.28658327244758523944484301805, −0.22455254233626395172840897537, 1.26895923756968390033750457884, 3.04884362141464010201349920565, 4.05647945269472244668975133884, 5.08971274255666185211525014547, 6.08289984694758459969232485256, 7.20293391657214458266907321140, 7.74708504646211554000628577761, 8.335616805136354895699425505802, 9.792931707349272615229294500533, 10.20678178656464657917379761757

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