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Properties

Label 2-867-51.5-c1-0-71
Degree $2$
Conductor $867$
Sign $-0.367 + 0.929i$
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  + (2.06 − 0.855i)2-s + (−0.290 − 1.70i)3-s + (2.12 − 2.12i)4-s + (2.77 + 0.551i)5-s + (−2.06 − 3.27i)6-s + (−0.616 − 3.10i)7-s + (0.855 − 2.06i)8-s + (−2.83 + 0.992i)9-s + (6.20 − 1.23i)10-s + (0.785 − 1.17i)11-s + (−4.23 − 3.00i)12-s + (2.82 + 2.82i)13-s + (−3.92 − 5.87i)14-s + (0.136 − 4.89i)15-s + 1.00i·16-s + ⋯
L(s)  = 1  + (1.46 − 0.605i)2-s + (−0.167 − 0.985i)3-s + (1.06 − 1.06i)4-s + (1.24 + 0.246i)5-s + (−0.841 − 1.33i)6-s + (−0.233 − 1.17i)7-s + (0.302 − 0.730i)8-s + (−0.943 + 0.330i)9-s + (1.96 − 0.390i)10-s + (0.236 − 0.354i)11-s + (−1.22 − 0.867i)12-s + (0.784 + 0.784i)13-s + (−1.04 − 1.57i)14-s + (0.0352 − 1.26i)15-s + 0.250i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14225 - 3.15018i\)
\(L(\frac12)\) \(\approx\) \(2.14225 - 3.15018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.290 + 1.70i)T \)
17 \( 1 \)
good2 \( 1 + (-2.06 + 0.855i)T + (1.41 - 1.41i)T^{2} \)
5 \( 1 + (-2.77 - 0.551i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (0.616 + 3.10i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-0.785 + 1.17i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-2.82 - 2.82i)T + 13iT^{2} \)
19 \( 1 + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.87 + 3.92i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.551 - 2.77i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (2.62 - 1.75i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (-3.51 - 5.25i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-5.54 + 1.10i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (-1.53 + 3.69i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (6.32 - 6.32i)T - 47iT^{2} \)
53 \( 1 + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.42 + 8.26i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (6.20 - 1.23i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + (10.5 - 7.07i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-2.62 - 1.75i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-3.42 - 8.26i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-9.48 - 9.48i)T + 89iT^{2} \)
97 \( 1 + (-12.4 - 2.46i)T + (89.6 + 37.1i)T^{2} \)
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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

−10.35343379982882896519801591813, −9.199321774472211980025231361132, −8.030361638294383961806635151351, −6.72660844450876178077952816038, −6.33795036701055217918228373763, −5.60970423921406774726408486845, −4.41440975953470410348220825309, −3.41407559296675764547298837353, −2.26673016505915888774157842462, −1.35660762143827021665202310315, 2.26734014720960238343897243703, 3.34089027393546407666876889823, 4.32679039078585450699240831802, 5.36631581553548573490224769006, 5.91031123111028979260601851363, 6.15340511414157849329561916208, 7.74350312527299138358058420114, 8.960596039812913205842424865843, 9.541300801990822138033033236848, 10.32411195503283287510307109887

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