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Properties

Label 2-867-17.15-c1-0-25
Degree $2$
Conductor $867$
Sign $0.0881 - 0.996i$
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 + (−0.382 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0881 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0881 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.0881 - 0.996i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.0881 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60310 + 2.38284i\)
\(L(\frac12)\) \(\approx\) \(2.60310 + 2.38284i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.65 + 1.10i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.19341916653446939167913889919, −9.382706114498259401014066884121, −8.269811481912892642591335818225, −7.58643294128379399789267908757, −6.54548435244510017770386981259, −5.98117638834943554851365171398, −5.28541098109855969988085106029, −4.57867081961417392173667914961, −3.13036319342334896584572484853, −1.77366780112375584168217479676, 1.42534426847571116049558740706, 2.68382852512108416869508814119, 3.24281276013034400297682232064, 4.62427612938746377309407670797, 5.29443160501143607311140176904, 6.07918367558172743521410662517, 6.82906192336235237992121183096, 8.748698469254580988613303763914, 9.656519264762943975126520768775, 10.32729345585817635541343355686

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