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Properties

Label 2-867-51.5-c1-0-39
Degree $2$
Conductor $867$
Sign $0.443 - 0.896i$
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  + (1.34 − 0.556i)2-s + (0.336 + 1.69i)3-s + (0.0830 − 0.0830i)4-s + (2.99 + 0.595i)5-s + (1.39 + 2.09i)6-s + (−0.420 − 2.11i)7-s + (−1.04 + 2.53i)8-s + (−2.77 + 1.14i)9-s + (4.35 − 0.866i)10-s + (−0.915 + 1.37i)11-s + (0.168 + 0.113i)12-s + (3.12 + 3.12i)13-s + (−1.74 − 2.60i)14-s + (−0.00349 + 5.28i)15-s + 4.22i·16-s + ⋯
L(s)  = 1  + (0.950 − 0.393i)2-s + (0.194 + 0.980i)3-s + (0.0415 − 0.0415i)4-s + (1.33 + 0.266i)5-s + (0.571 + 0.855i)6-s + (−0.159 − 0.799i)7-s + (−0.370 + 0.894i)8-s + (−0.924 + 0.381i)9-s + (1.37 − 0.273i)10-s + (−0.276 + 0.413i)11-s + (0.0487 + 0.0326i)12-s + (0.866 + 0.866i)13-s + (−0.465 − 0.697i)14-s + (−0.000903 + 1.36i)15-s + 1.05i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.443 - 0.896i$
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: \((2,\ 867,\ (\ :1/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50355 + 1.55497i\)
\(L(\frac12)\) \(\approx\) \(2.50355 + 1.55497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.336 - 1.69i)T \)
17 \( 1 \)
good2 \( 1 + (-1.34 + 0.556i)T + (1.41 - 1.41i)T^{2} \)
5 \( 1 + (-2.99 - 0.595i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (0.420 + 2.11i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.915 - 1.37i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-3.12 - 3.12i)T + 13iT^{2} \)
19 \( 1 + (-0.330 - 0.797i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.637 + 0.425i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-1.55 + 7.83i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (-2.40 + 1.60i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (-1.32 - 1.98i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-3.02 + 0.600i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (4.38 - 10.5i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (-2.21 + 2.21i)T - 47iT^{2} \)
53 \( 1 + (5.88 - 2.43i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.33 + 5.62i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.73 + 1.14i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 - 7.19iT - 67T^{2} \)
71 \( 1 + (1.80 - 1.20i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-2.45 + 12.3i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (4.95 + 3.31i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (4.96 + 11.9i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (3.42 + 3.42i)T + 89iT^{2} \)
97 \( 1 + (-3.47 - 0.692i)T + (89.6 + 37.1i)T^{2} \)
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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

−10.22179837614281392262574537286, −9.740191763806832623574136459538, −8.865209046369155570200496209155, −7.895779117633110389679820952214, −6.39724159862003106554612889419, −5.79602510600683810671197640514, −4.68555872552062875550872073366, −4.09086455425154617313098338216, −3.04179434589093342398657181379, −2.03414136797447177213099689626, 1.11514335319191800547362864467, 2.49604306832822513373489170852, 3.45781031706202169118976693275, 5.18156778330156490233683849147, 5.68597001575030010194645889756, 6.21410129480672955617074690177, 7.09652726348147102313785496110, 8.446723313537709353389487963285, 8.975487024798659752578536657046, 9.896526506471305487436192746112

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