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Properties

Label 2-867-17.8-c1-0-2
Degree $2$
Conductor $867$
Sign $-0.940 - 0.339i$
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  + (0.707 − 0.707i)2-s + (−0.382 + 0.923i)3-s + 0.999i·4-s + (0.382 + 0.923i)6-s + (−3.69 + 1.53i)7-s + (2.12 + 2.12i)8-s + (−0.707 − 0.707i)9-s + (−1.53 − 3.69i)11-s + (−0.923 − 0.382i)12-s + 2i·13-s + (−1.53 + 3.69i)14-s + 1.00·16-s − 18-s + (−2.82 + 2.82i)19-s − 4i·21-s + (−3.69 − 1.53i)22-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.220 + 0.533i)3-s + 0.499i·4-s + (0.156 + 0.377i)6-s + (−1.39 + 0.578i)7-s + (0.750 + 0.750i)8-s + (−0.235 − 0.235i)9-s + (−0.461 − 1.11i)11-s + (−0.266 − 0.110i)12-s + 0.554i·13-s + (−0.409 + 0.987i)14-s + 0.250·16-s − 0.235·18-s + (−0.648 + 0.648i)19-s − 0.872i·21-s + (−0.787 − 0.326i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.940 - 0.339i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (688, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104049 + 0.595136i\)
\(L(\frac12)\) \(\approx\) \(0.104049 + 0.595136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \)
5 \( 1 + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (3.69 - 1.53i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.53 + 3.69i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
19 \( 1 + (2.82 - 2.82i)T - 19iT^{2} \)
23 \( 1 + (1.53 + 3.69i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.53 - 3.69i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (3.06 - 7.39i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.39 - 3.06i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.82 + 2.82i)T + 43iT^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + (-8.48 - 8.48i)T + 59iT^{2} \)
61 \( 1 + (-7.39 + 3.06i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (4.59 - 11.0i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.53 - 3.69i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (-14.7 - 6.12i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.41252025180402858872656821188, −10.02946820164429549419264040936, −8.661804385085212159818017361003, −8.419534821041530244724118059994, −6.88244070622467860469293816263, −6.08666585440264035361801425179, −5.15529153109948006862196402932, −3.95710732473710526284402128925, −3.30612688028365868678210178079, −2.34766066978272807545617520910, 0.23517604081188718018490230396, 1.94277178718716725005421179908, 3.45185909883468290310077739643, 4.54016092189191107059059238212, 5.58859706278923722672319078704, 6.26961128740978706507298897125, 7.20118741517240598563087388531, 7.50826268639809336915994343928, 9.085019630689094729743167180522, 9.974908507053732155939682591729

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