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Properties

Label 2-867-17.2-c1-0-12
Degree $2$
Conductor $867$
Sign $0.892 + 0.451i$
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 + (0.923 − 0.382i)3-s − 0.999i·4-s + (−0.923 − 0.382i)6-s + (−1.53 + 3.69i)7-s + (−2.12 + 2.12i)8-s + (0.707 − 0.707i)9-s + (3.69 + 1.53i)11-s + (−0.382 − 0.923i)12-s − 2i·13-s + (3.69 − 1.53i)14-s + 1.00·16-s − 18-s + (2.82 + 2.82i)19-s + 4i·21-s + (−1.53 − 3.69i)22-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.533 − 0.220i)3-s − 0.499i·4-s + (−0.377 − 0.156i)6-s + (−0.578 + 1.39i)7-s + (−0.750 + 0.750i)8-s + (0.235 − 0.235i)9-s + (1.11 + 0.461i)11-s + (−0.110 − 0.266i)12-s − 0.554i·13-s + (0.987 − 0.409i)14-s + 0.250·16-s − 0.235·18-s + (0.648 + 0.648i)19-s + 0.872i·21-s + (−0.326 − 0.787i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.892 + 0.451i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.892 + 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36922 - 0.326854i\)
\(L(\frac12)\) \(\approx\) \(1.36922 - 0.326854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−9.747328095169653495231550667158, −9.393047344998711852119862930814, −8.727607442496370607302063692032, −7.80791904070714827148830182802, −6.47455019664446195319623229768, −5.92753230680718389690882965608, −4.78606879705595524514335723458, −3.22511975577162729177830960956, −2.42039625479470114802521420114, −1.20483173969713385711860296478, 0.945405578125172076535186789652, 3.03528947557504314299614307156, 3.73523041377793760460188537231, 4.62835626845921216495864560829, 6.36407251010100144758003164976, 6.98385948811711364029101867839, 7.55913867595604050795332049598, 8.675283600805957015524845243132, 9.205113436857697681937477501400, 9.915354717436962896678026941693

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