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Properties

Label 2-867-17.4-c1-0-40
Degree $2$
Conductor $867$
Sign $-0.591 - 0.805i$
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  − 2.56i·2-s + (0.707 − 0.707i)3-s − 4.56·4-s + (2.51 − 2.51i)5-s + (−1.81 − 1.81i)6-s + 6.56i·8-s − 1.00i·9-s + (−6.45 − 6.45i)10-s + (−1.10 − 1.10i)11-s + (−3.22 + 3.22i)12-s − 0.438·13-s − 3.56i·15-s + 7.68·16-s − 2.56·18-s − 4.68i·19-s + (−11.4 + 11.4i)20-s + ⋯
L(s)  = 1  − 1.81i·2-s + (0.408 − 0.408i)3-s − 2.28·4-s + (1.12 − 1.12i)5-s + (−0.739 − 0.739i)6-s + 2.31i·8-s − 0.333i·9-s + (−2.03 − 2.03i)10-s + (−0.332 − 0.332i)11-s + (−0.931 + 0.931i)12-s − 0.121·13-s − 0.919i·15-s + 1.92·16-s − 0.603·18-s − 1.07i·19-s + (−2.56 + 2.56i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.591 - 0.805i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (616, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.591 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.754237 + 1.48976i\)
\(L(\frac12)\) \(\approx\) \(0.754237 + 1.48976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 \)
good2 \( 1 + 2.56iT - 2T^{2} \)
5 \( 1 + (-2.51 + 2.51i)T - 5iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + (1.10 + 1.10i)T + 11iT^{2} \)
13 \( 1 + 0.438T + 13T^{2} \)
19 \( 1 + 4.68iT - 19T^{2} \)
23 \( 1 + (-1.72 - 1.72i)T + 23iT^{2} \)
29 \( 1 + (5.83 - 5.83i)T - 29iT^{2} \)
31 \( 1 + (2.20 - 2.20i)T - 31iT^{2} \)
37 \( 1 + (-3.62 + 3.62i)T - 37iT^{2} \)
41 \( 1 + (2.51 + 2.51i)T + 41iT^{2} \)
43 \( 1 + 4.68iT - 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 7.12iT - 59T^{2} \)
61 \( 1 + (-6.45 - 6.45i)T + 61iT^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-4.41 + 4.41i)T - 71iT^{2} \)
73 \( 1 + (8.65 - 8.65i)T - 73iT^{2} \)
79 \( 1 + (-6.62 - 6.62i)T + 79iT^{2} \)
83 \( 1 + 0.876iT - 83T^{2} \)
89 \( 1 - 1.12T + 89T^{2} \)
97 \( 1 + (2.03 - 2.03i)T - 97iT^{2} \)
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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

−9.568110660554259069769815911765, −9.033885049188348456750659064195, −8.583481762509458199638944298464, −7.20102699325439246059027015828, −5.62295797917287848129302545041, −5.00820725990604398232818057388, −3.83915131936647229977628940161, −2.66229164674751554832406309643, −1.83192084429667366178153621055, −0.78870966170490182508904131112, 2.29884158436761615781354291535, 3.67981086009991759012062681261, 4.83410296559012718490640307766, 5.82038750314049951411770921492, 6.30376453174978846283219152266, 7.30921963894506587681727286649, 7.88496958986249864162673440898, 8.923107057491690318872790280855, 9.771670895413654727862089432792, 10.13172324982374613126842616377

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