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Properties

Label 2-867-51.2-c0-0-0
Degree $2$
Conductor $867$
Sign $-0.766 - 0.641i$
Analytic cond. $0.432689$
Root an. cond. $0.657791$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)3-s + i·4-s + (−0.382 + 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.382 − 0.923i)12-s + i·13-s − 16-s + (−0.707 − 0.707i)19-s − i·21-s + (−0.707 + 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 − 0.382i)28-s + (−0.923 + 0.382i)31-s + (0.707 + 0.707i)36-s + (0.923 − 0.382i)37-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + i·4-s + (−0.382 + 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.382 − 0.923i)12-s + i·13-s − 16-s + (−0.707 − 0.707i)19-s − i·21-s + (−0.707 + 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.923 − 0.382i)28-s + (−0.923 + 0.382i)31-s + (0.707 + 0.707i)36-s + (0.923 − 0.382i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.766 - 0.641i$
Analytic conductor: \(0.432689\)
Root analytic conductor: \(0.657791\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :0),\ -0.766 - 0.641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5682174750\)
\(L(\frac12)\) \(\approx\) \(0.5682174750\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.382 - 0.923i)T + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 - iT - T^{2} \)
19 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-0.382 + 0.923i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.765 - 1.84i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1.84 - 0.765i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.382 - 0.923i)T + (-0.707 + 0.707i)T^{2} \)
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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

−11.06258913277426156903390596079, −9.607366004312563640444562770179, −9.209344989691559448414941927491, −8.243860437815478804764483796257, −7.06869835139934465974449310806, −6.45043205844592216306177969425, −5.42174537559602722660763293303, −4.42263925596450581053736937023, −3.53577873862173420936528782757, −2.20549631315160500851622577935, 0.62625988479081856272552325485, 2.02142271067566542492406687942, 3.84051442890446429806688923347, 4.87545669220120192063998367424, 5.81117671675278552320785403460, 6.39654289074764816085500418947, 7.30459150814530236122024310263, 8.168009898068231008483121569042, 9.555554697214944076824948565018, 10.33272846224804028859234870005

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