↑

Properties

Label 2-867-17.2-c1-0-21
Degree $2$
Conductor $867$
Sign $0.692 - 0.721i$
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  + (1.10 + 1.10i)2-s + (−0.923 + 0.382i)3-s + 0.438i·4-s + (0.214 + 0.518i)5-s + (−1.44 − 0.597i)6-s + (1.72 − 1.72i)8-s + (0.707 − 0.707i)9-s + (−0.335 + 0.810i)10-s + (2.36 + 0.980i)11-s + (−0.167 − 0.405i)12-s − 4.56i·13-s + (−0.397 − 0.397i)15-s + 4.68·16-s + 1.56·18-s + (5.43 + 5.43i)19-s + (−0.227 + 0.0942i)20-s + ⋯
L(s)  = 1  + (0.780 + 0.780i)2-s + (−0.533 + 0.220i)3-s + 0.219i·4-s + (0.0961 + 0.232i)5-s + (−0.588 − 0.243i)6-s + (0.609 − 0.609i)8-s + (0.235 − 0.235i)9-s + (−0.106 + 0.256i)10-s + (0.713 + 0.295i)11-s + (−0.0484 − 0.116i)12-s − 1.26i·13-s + (−0.102 − 0.102i)15-s + 1.17·16-s + 0.368·18-s + (1.24 + 1.24i)19-s + (−0.0508 + 0.0210i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.692 - 0.721i$
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: \((2,\ 867,\ (\ :1/2),\ 0.692 - 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06249 + 0.878850i\)
\(L(\frac12)\) \(\approx\) \(2.06249 + 0.878850i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 - 1.10i)T + 2iT^{2} \)
5 \( 1 + (-0.214 - 0.518i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.36 - 0.980i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.56iT - 13T^{2} \)
19 \( 1 + (-5.43 - 5.43i)T + 19iT^{2} \)
23 \( 1 + (6.06 + 2.51i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.15 - 7.61i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-4.73 + 1.96i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-2.88 + 1.19i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.214 - 0.518i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.43 - 5.43i)T - 43iT^{2} \)
47 \( 1 - 2.87iT - 47T^{2} \)
53 \( 1 + (3.00 + 3.00i)T + 53iT^{2} \)
59 \( 1 + (-0.794 + 0.794i)T - 59iT^{2} \)
61 \( 1 + (-0.335 + 0.810i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-9.46 + 3.92i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.62 + 3.92i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (14.1 + 5.88i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-6.45 - 6.45i)T + 83iT^{2} \)
89 \( 1 - 7.12iT - 89T^{2} \)
97 \( 1 + (4.25 + 10.2i)T + (-68.5 + 68.5i)T^{2} \)
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

−10.15031663603685535475124785718, −9.775125171966670309893985203424, −8.296457461687856595175004443632, −7.48956473681559436653353745606, −6.49678941973416741758131975606, −5.94405028733250701452861255604, −5.10265602101812621148954532218, −4.23220701133546769353362079814, −3.17134552351130430861017733877, −1.20042666886859258081775370893, 1.27859468303372649660446279256, 2.48994525867710573105251416010, 3.74685538563202013268917452246, 4.56356332364212239871482564756, 5.40598675536976289457832248192, 6.48821609431263846601664247399, 7.33442703889859661876426868136, 8.421502241924301965486005416815, 9.378091294991198235728384091340, 10.23476570701381057039183048942

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