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Properties

Label 2-867-17.15-c1-0-38
Degree $2$
Conductor $867$
Sign $-0.691 + 0.722i$
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  + (−0.157 − 0.157i)2-s + (−0.382 − 0.923i)3-s − 1.95i·4-s + (0.587 − 0.243i)5-s + (−0.0850 + 0.205i)6-s + (1.59 + 0.661i)7-s + (−0.620 + 0.620i)8-s + (−0.707 + 0.707i)9-s + (−0.130 − 0.0541i)10-s + (1.89 − 4.57i)11-s + (−1.80 + 0.746i)12-s − 2.50i·13-s + (−0.146 − 0.354i)14-s + (−0.450 − 0.450i)15-s − 3.70·16-s + ⋯
L(s)  = 1  + (−0.111 − 0.111i)2-s + (−0.220 − 0.533i)3-s − 0.975i·4-s + (0.262 − 0.108i)5-s + (−0.0347 + 0.0838i)6-s + (0.603 + 0.250i)7-s + (−0.219 + 0.219i)8-s + (−0.235 + 0.235i)9-s + (−0.0413 − 0.0171i)10-s + (0.571 − 1.37i)11-s + (−0.520 + 0.215i)12-s − 0.695i·13-s + (−0.0392 − 0.0948i)14-s + (−0.116 − 0.116i)15-s − 0.926·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.542843 - 1.27134i\)
\(L(\frac12)\) \(\approx\) \(0.542843 - 1.27134i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.157 + 0.157i)T + 2iT^{2} \)
5 \( 1 + (-0.587 + 0.243i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.59 - 0.661i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.89 + 4.57i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 2.50iT - 13T^{2} \)
19 \( 1 + (-0.672 - 0.672i)T + 19iT^{2} \)
23 \( 1 + (-0.812 + 1.96i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-8.85 + 3.66i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.01 - 4.87i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.24 + 7.84i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (6.39 + 2.64i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (5.05 - 5.05i)T - 43iT^{2} \)
47 \( 1 + 8.10iT - 47T^{2} \)
53 \( 1 + (-4.55 - 4.55i)T + 53iT^{2} \)
59 \( 1 + (7.15 - 7.15i)T - 59iT^{2} \)
61 \( 1 + (-2.74 - 1.13i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 7.70T + 67T^{2} \)
71 \( 1 + (-0.146 - 0.354i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (6.01 - 2.49i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.908 - 2.19i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (9.44 + 9.44i)T + 83iT^{2} \)
89 \( 1 + 1.98iT - 89T^{2} \)
97 \( 1 + (-2.80 + 1.16i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.05199806128222807686038505684, −8.781134101094659457268203098614, −8.458838757337800043338353534397, −7.17481993773744874272160362273, −6.13972912478137993132237049478, −5.65410183983885333378524335385, −4.74081053219576446929172618860, −3.14908497524001693845625511946, −1.80162700834356188553045164179, −0.74049081023803824326266836457, 1.82384264091108533017572233655, 3.18736683820128230865549053291, 4.39154532073829220262490702729, 4.78104079102333873627113625601, 6.38678393683933334591795045462, 7.02569628655122764636812565909, 7.977524914601797377965610615894, 8.792067206380493714955344175233, 9.680426659546371118490902397742, 10.27951457947219066914205106452

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