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Properties

Label 2-867-17.2-c1-0-0
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.81 − 1.81i)2-s + (−0.923 + 0.382i)3-s + 4.56i·4-s + (−1.36 − 3.29i)5-s + (2.36 + 0.980i)6-s + (4.63 − 4.63i)8-s + (0.707 − 0.707i)9-s + (−3.49 + 8.42i)10-s + (−1.44 − 0.597i)11-s + (−1.74 − 4.21i)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 + (15.0 − 6.21i)20-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)2-s + (−0.533 + 0.220i)3-s + 2.28i·4-s + (−0.609 − 1.47i)5-s + (0.966 + 0.400i)6-s + (1.64 − 1.64i)8-s + (0.235 − 0.235i)9-s + (−1.10 + 2.66i)10-s + (−0.434 − 0.180i)11-s + (−0.503 − 1.21i)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 + (3.35 − 1.39i)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

\(L(1)\) \(\approx\) \(0.00192872 + 0.000821850i\)
\(L(\frac12)\) \(\approx\) \(0.00192872 + 0.000821850i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 + 1.81i)T + 2iT^{2} \)
5 \( 1 + (1.36 + 3.29i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.44 + 0.597i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 0.438iT - 13T^{2} \)
19 \( 1 + (3.31 + 3.31i)T + 19iT^{2} \)
23 \( 1 + (2.25 + 0.933i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.15 + 7.61i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.88 - 1.19i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (4.73 - 1.96i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.36 + 3.29i)T + (-28.9 - 28.9i)T^{2} \)
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 \( 1 + (-3.49 + 8.42i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (5.77 - 2.39i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-4.68 - 11.3i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-8.65 - 3.58i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.620 - 0.620i)T + 83iT^{2} \)
89 \( 1 + 1.12iT - 89T^{2} \)
97 \( 1 + (1.10 + 2.65i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.34554257165235663963459182409, −9.364808188040515202987428617857, −8.858236097024748492079282357790, −8.108465074639081535022957141116, −7.36553989209143709494134161650, −5.77488039138662706520792915617, −4.57168448564369644871911123726, −3.85572382997256971496125701747, −2.38403640328456614858947613607, −1.00085534401649247408981065753, 0.00206598616757008176641321448, 1.97942600396342935905713393779, 3.69268395575779235998565643569, 5.24944194558493785281232567963, 6.12215756041623919280137005143, 6.88703377956346796747460015617, 7.37642254389651950875987537940, 8.091935124568628807741803558116, 9.027559050617832641818994147785, 10.22451684621842863387062512285

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