↑

Properties

Label 2-867-17.8-c1-0-45
Degree $2$
Conductor $867$
Sign $-0.750 - 0.660i$
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  + (1.81 − 1.81i)2-s + (0.382 − 0.923i)3-s − 4.56i·4-s + (−3.29 − 1.36i)5-s + (−0.980 − 2.36i)6-s + (−4.63 − 4.63i)8-s + (−0.707 − 0.707i)9-s + (−8.42 + 3.49i)10-s + (0.597 + 1.44i)11-s + (−4.21 − 1.74i)12-s + 0.438i·13-s + (−2.51 + 2.51i)15-s − 7.68·16-s − 2.56·18-s + (3.31 − 3.31i)19-s + (−6.21 + 15.0i)20-s + ⋯
L(s)  = 1  + (1.28 − 1.28i)2-s + (0.220 − 0.533i)3-s − 2.28i·4-s + (−1.47 − 0.609i)5-s + (−0.400 − 0.966i)6-s + (−1.64 − 1.64i)8-s + (−0.235 − 0.235i)9-s + (−2.66 + 1.10i)10-s + (0.180 + 0.434i)11-s + (−1.21 − 0.503i)12-s + 0.121i·13-s + (−0.650 + 0.650i)15-s − 1.92·16-s − 0.603·18-s + (0.759 − 0.759i)19-s + (−1.39 + 3.35i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.750 - 0.660i$
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: \((2,\ 867,\ (\ :1/2),\ -0.750 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757026 + 2.00581i\)
\(L(\frac12)\) \(\approx\) \(0.757026 + 2.00581i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 1.81i)T - 2iT^{2} \)
5 \( 1 + (3.29 + 1.36i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.597 - 1.44i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 0.438iT - 13T^{2} \)
19 \( 1 + (-3.31 + 3.31i)T - 19iT^{2} \)
23 \( 1 + (-0.933 - 2.25i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (7.61 + 3.15i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.19 + 2.88i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.96 + 4.73i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-3.29 + 1.36i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (3.31 + 3.31i)T + 43iT^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 + (8.65 - 8.65i)T - 53iT^{2} \)
59 \( 1 + (-5.03 - 5.03i)T + 59iT^{2} \)
61 \( 1 + (-8.42 + 3.49i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-2.39 + 5.77i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-11.3 - 4.68i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.58 + 8.65i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (0.620 - 0.620i)T - 83iT^{2} \)
89 \( 1 - 1.12iT - 89T^{2} \)
97 \( 1 + (2.65 + 1.10i)T + (68.5 + 68.5i)T^{2} \)
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

−9.770380025421790496265724467442, −9.010292891012436839141435844457, −7.82861144252545397339028609383, −7.08295957246280240538629898219, −5.75283896040336432255854553665, −4.81023436042222630387828238812, −4.01316775967871790647930559604, −3.28360464815967888885707128000, −2.02240771998267937032846080967, −0.67423871272799788294863248285, 3.12871371280771715206270827850, 3.57057589002874204818373029215, 4.44892409561905685475486946180, 5.31874939051413367824057901052, 6.37721445776640230683774761374, 7.17543774931723103718054193593, 7.964130175205772935570374675690, 8.412525874548706015904862757594, 9.725298336038569323660693150237, 11.06539955078114013448904030124

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