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Properties

Label 2-867-17.8-c1-0-42
Degree $2$
Conductor $867$
Sign $-0.774 + 0.632i$
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  + (1.86 − 1.86i)2-s + (0.382 − 0.923i)3-s − 4.95i·4-s + (2.05 + 0.850i)5-s + (−1.00 − 2.43i)6-s + (2.13 − 0.885i)7-s + (−5.50 − 5.50i)8-s + (−0.707 − 0.707i)9-s + (5.41 − 2.24i)10-s + (0.746 + 1.80i)11-s + (−4.57 − 1.89i)12-s + 1.32i·13-s + (2.33 − 5.63i)14-s + (1.57 − 1.57i)15-s − 10.6·16-s + ⋯
L(s)  = 1  + (1.31 − 1.31i)2-s + (0.220 − 0.533i)3-s − 2.47i·4-s + (0.918 + 0.380i)5-s + (−0.411 − 0.994i)6-s + (0.808 − 0.334i)7-s + (−1.94 − 1.94i)8-s + (−0.235 − 0.235i)9-s + (1.71 − 0.708i)10-s + (0.225 + 0.543i)11-s + (−1.32 − 0.546i)12-s + 0.366i·13-s + (0.624 − 1.50i)14-s + (0.405 − 0.405i)15-s − 2.65·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.774 + 0.632i$
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: \((2,\ 867,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33322 - 3.74152i\)
\(L(\frac12)\) \(\approx\) \(1.33322 - 3.74152i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 1.86i)T - 2iT^{2} \)
5 \( 1 + (-2.05 - 0.850i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-2.13 + 0.885i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.746 - 1.80i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.32iT - 13T^{2} \)
19 \( 1 + (4.20 - 4.20i)T - 19iT^{2} \)
23 \( 1 + (-1.82 - 4.41i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.159 - 0.0661i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.170 + 0.410i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (2.03 - 4.91i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.71 - 1.53i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.89 + 5.89i)T + 43iT^{2} \)
47 \( 1 - 2.38iT - 47T^{2} \)
53 \( 1 + (-0.514 + 0.514i)T - 53iT^{2} \)
59 \( 1 + (8.33 + 8.33i)T + 59iT^{2} \)
61 \( 1 + (8.02 - 3.32i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + (2.33 - 5.63i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-9.75 - 4.03i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-0.267 - 0.645i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-6.03 + 6.03i)T - 83iT^{2} \)
89 \( 1 + 5.40iT - 89T^{2} \)
97 \( 1 + (4.35 + 1.80i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.10584205324034078540211526630, −9.456885166489806479624457261137, −8.185210198374708319334375631172, −6.87239481762771400980575037345, −6.11779949563520960301691120981, −5.19151641326648603315303964567, −4.28382250873830415738567860482, −3.24762655160821314057724715100, −1.99663720579560626068640458967, −1.58230030871075930494426176483, 2.30688924426656809789824795720, 3.49459531742427532077297799383, 4.67859222617654356443760316803, 5.11262328735080520786306830329, 5.99216364363408334673388001455, 6.71989472833708735117996018842, 7.921378633594603458007697386524, 8.607594534554913374347951246629, 9.224585084761111369459976383165, 10.59300299150374421437219314683

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