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Properties

Label 2-867-17.15-c1-0-21
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  + (−1.41 − 1.41i)2-s + (−0.382 − 0.923i)3-s + 2.00i·4-s + (−2.77 + 1.14i)5-s + (−0.765 + 1.84i)6-s + (1.84 + 0.765i)7-s + (−0.707 + 0.707i)9-s + (5.54 + 2.29i)10-s + (1.91 − 4.61i)11-s + (1.84 − 0.765i)12-s + i·13-s + (−1.53 − 3.69i)14-s + (2.12 + 2.12i)15-s + 3.99·16-s + 2·18-s + (3.53 + 3.53i)19-s + ⋯
L(s)  = 1  + (−0.999 − 0.999i)2-s + (−0.220 − 0.533i)3-s + 1.00i·4-s + (−1.23 + 0.513i)5-s + (−0.312 + 0.754i)6-s + (0.698 + 0.289i)7-s + (−0.235 + 0.235i)9-s + (1.75 + 0.726i)10-s + (0.576 − 1.39i)11-s + (0.533 − 0.220i)12-s + 0.277i·13-s + (−0.409 − 0.987i)14-s + (0.547 + 0.547i)15-s + 0.999·16-s + 0.471·18-s + (0.811 + 0.811i)19-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: $\chi_{867} (712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0909091 - 0.519974i\)
\(L(\frac12)\) \(\approx\) \(0.0909091 - 0.519974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 1.41i)T + 2iT^{2} \)
5 \( 1 + (2.77 - 1.14i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.84 - 0.765i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.91 + 4.61i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
19 \( 1 + (-3.53 - 3.53i)T + 19iT^{2} \)
23 \( 1 + (0.382 - 0.923i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-5.54 + 2.29i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (3.82 + 9.23i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.765 + 1.84i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.61 + 1.91i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - 43iT^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (9.23 + 3.82i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.54 + 2.29i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.53 + 3.69i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 + (7.39 - 3.06i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.862553497052123121934326718937, −8.900517893190871455933020505897, −8.063861488901506931108634010755, −7.77349720152684946652203040098, −6.46773849176523528087557971064, −5.46193282823251678570633406771, −3.88464749561289881567933701158, −3.06473020925539017408501389026, −1.73129276358341186624772577399, −0.45580765796521759949327945374, 1.13598270452360027363626434685, 3.42697979692763419976340983876, 4.56818036343149690195972046759, 5.12084711150747418311491842537, 6.66379319609701375240194650983, 7.26512148021002329828992499959, 8.025711885124497113139761339912, 8.708288138011500398177045007234, 9.455850625382559577652808879821, 10.29293717999168618020054587667

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