↑

Properties

Label 2-867-51.41-c1-0-43
Degree $2$
Conductor $867$
Sign $-0.112 + 0.993i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 − 0.556i)2-s + (−1.44 + 0.961i)3-s + (0.0830 + 0.0830i)4-s + (2.99 − 0.595i)5-s + (2.47 − 0.490i)6-s + (0.420 − 2.11i)7-s + (1.04 + 2.53i)8-s + (1.15 − 2.77i)9-s + (−4.35 − 0.866i)10-s + (−0.915 − 1.37i)11-s + (−0.199 − 0.0397i)12-s + (3.12 − 3.12i)13-s + (−1.74 + 2.60i)14-s + (−3.73 + 3.73i)15-s − 4.22i·16-s + ⋯
L(s)  = 1  + (−0.950 − 0.393i)2-s + (−0.831 + 0.555i)3-s + (0.0415 + 0.0415i)4-s + (1.33 − 0.266i)5-s + (1.00 − 0.200i)6-s + (0.159 − 0.799i)7-s + (0.370 + 0.894i)8-s + (0.383 − 0.923i)9-s + (−1.37 − 0.273i)10-s + (−0.276 − 0.413i)11-s + (−0.0575 − 0.0114i)12-s + (0.866 − 0.866i)13-s + (−0.465 + 0.697i)14-s + (−0.965 + 0.964i)15-s − 1.05i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.112 + 0.993i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.112 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529411 - 0.592912i\)
\(L(\frac12)\) \(\approx\) \(0.529411 - 0.592912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.44 - 0.961i)T \)
17 \( 1 \)
good2 \( 1 + (1.34 + 0.556i)T + (1.41 + 1.41i)T^{2} \)
5 \( 1 + (-2.99 + 0.595i)T + (4.61 - 1.91i)T^{2} \)
7 \( 1 + (-0.420 + 2.11i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (0.915 + 1.37i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (-3.12 + 3.12i)T - 13iT^{2} \)
19 \( 1 + (-0.330 + 0.797i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.637 - 0.425i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.55 - 7.83i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (2.40 + 1.60i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (1.32 - 1.98i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-3.02 - 0.600i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (4.38 + 10.5i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (2.21 + 2.21i)T + 47iT^{2} \)
53 \( 1 + (-5.88 - 2.43i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.33 + 5.62i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (5.73 + 1.14i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + 7.19iT - 67T^{2} \)
71 \( 1 + (1.80 + 1.20i)T + (27.1 + 65.5i)T^{2} \)
73 \( 1 + (2.45 + 12.3i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-4.95 + 3.31i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (-4.96 + 11.9i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-3.42 + 3.42i)T - 89iT^{2} \)
97 \( 1 + (3.47 - 0.692i)T + (89.6 - 37.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.17434086004128302363853329279, −9.232428497504554631451573115386, −8.687213944830822091373123860095, −7.50360832678934970811489811822, −6.27417608135547119100585232973, −5.49726172712834885495716016171, −4.85160895425082057058110316355, −3.40050794646185829998768782938, −1.69149419743971388875413588562, −0.66115866459389541103170479053, 1.38009026420730149662392008310, 2.36294379799962628617285156864, 4.30251263603742788379467560202, 5.53479869553471274566988190403, 6.21963508911636641293902295538, 6.86857711348076259878896683036, 7.88219247518759270772913634259, 8.720974030672276654948580182973, 9.574623715065504403083539730791, 10.13509128643820352262626078204

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