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Properties

Label 2-867-17.9-c1-0-1
Degree $2$
Conductor $867$
Sign $-0.557 + 0.830i$
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  + (−0.541 + 0.541i)2-s + (−0.923 − 0.382i)3-s + 1.41i·4-s + (−0.796 + 1.92i)5-s + (0.707 − 0.292i)6-s + (−0.472 − 1.14i)7-s + (−1.84 − 1.84i)8-s + (0.707 + 0.707i)9-s + (−0.609 − 1.47i)10-s + (1.38 − 0.572i)11-s + (0.541 − 1.30i)12-s + 4.10i·13-s + (0.873 + 0.361i)14-s + (1.47 − 1.47i)15-s − 0.828·16-s + ⋯
L(s)  = 1  + (−0.382 + 0.382i)2-s + (−0.533 − 0.220i)3-s + 0.707i·4-s + (−0.356 + 0.860i)5-s + (0.288 − 0.119i)6-s + (−0.178 − 0.431i)7-s + (−0.653 − 0.653i)8-s + (0.235 + 0.235i)9-s + (−0.192 − 0.465i)10-s + (0.416 − 0.172i)11-s + (0.156 − 0.377i)12-s + 1.13i·13-s + (0.233 + 0.0966i)14-s + (0.380 − 0.380i)15-s − 0.207·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.557 + 0.830i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0711974 - 0.133537i\)
\(L(\frac12)\) \(\approx\) \(0.0711974 - 0.133537i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.541 - 0.541i)T - 2iT^{2} \)
5 \( 1 + (0.796 - 1.92i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.472 + 1.14i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.38 + 0.572i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
19 \( 1 + (4.81 - 4.81i)T - 19iT^{2} \)
23 \( 1 + (3.05 - 1.26i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.38 + 5.76i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (9.48 + 3.93i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-3.63 - 1.50i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.64 + 11.2i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (2.43 + 2.43i)T + 43iT^{2} \)
47 \( 1 - 1.56iT - 47T^{2} \)
53 \( 1 + (-2.80 + 2.80i)T - 53iT^{2} \)
59 \( 1 + (5.70 + 5.70i)T + 59iT^{2} \)
61 \( 1 + (-1.25 - 3.02i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 + (0.209 + 0.0867i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.140 + 0.340i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (2.50 - 1.03i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + (1.02 - 2.48i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.74392702360577788562632645183, −9.870315717489265326539162429978, −8.915683261640536283896518775320, −8.005518210588683338042826377981, −7.18697767271624475951205450593, −6.67664764679782532661538008693, −5.86696977358365416046026988209, −4.12957203640395257065112036833, −3.67336406573004093312240149275, −2.09659581753551033562299797315, 0.091722092717151946573434336817, 1.37268696683713424265506941589, 2.85677233651254276039031233741, 4.40046869322715915421429007305, 5.15353647429860409569096929841, 5.96134800584207828892077846853, 6.88384752215525829265814670751, 8.283439305131867788466859793429, 8.902641450832270577445685667151, 9.609644609877069573255957572297

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