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Properties

Label 2-867-17.16-c1-0-14
Degree $2$
Conductor $867$
Sign $-0.685 - 0.727i$
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  + 1.65·2-s + i·3-s + 0.729·4-s + 4.06i·5-s + 1.65i·6-s + 0.922i·7-s − 2.09·8-s − 9-s + 6.71i·10-s − 2.27i·11-s + 0.729i·12-s + 3.57·13-s + 1.52i·14-s − 4.06·15-s − 4.92·16-s + ⋯
L(s)  = 1  + 1.16·2-s + 0.577i·3-s + 0.364·4-s + 1.81i·5-s + 0.674i·6-s + 0.348i·7-s − 0.741·8-s − 0.333·9-s + 2.12i·10-s − 0.684i·11-s + 0.210i·12-s + 0.991·13-s + 0.407i·14-s − 1.04·15-s − 1.23·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.685 - 0.727i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884960 + 2.05061i\)
\(L(\frac12)\) \(\approx\) \(0.884960 + 2.05061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
17 \( 1 \)
good2 \( 1 - 1.65T + 2T^{2} \)
5 \( 1 - 4.06iT - 5T^{2} \)
7 \( 1 - 0.922iT - 7T^{2} \)
11 \( 1 + 2.27iT - 11T^{2} \)
13 \( 1 - 3.57T + 13T^{2} \)
19 \( 1 + 1.72T + 19T^{2} \)
23 \( 1 - 5.09iT - 23T^{2} \)
29 \( 1 - 2.20iT - 29T^{2} \)
31 \( 1 + 4.13iT - 31T^{2} \)
37 \( 1 - 5.95iT - 37T^{2} \)
41 \( 1 - 5.05iT - 41T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 + 6.96T + 47T^{2} \)
53 \( 1 - 2.69T + 53T^{2} \)
59 \( 1 - 7.39T + 59T^{2} \)
61 \( 1 + 3.89iT - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 1.52iT - 71T^{2} \)
73 \( 1 + 1.16iT - 73T^{2} \)
79 \( 1 + 11.8iT - 79T^{2} \)
83 \( 1 - 6.38T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 14.4iT - 97T^{2} \)
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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

−10.71815788996985103733533906076, −9.801545897500303119054883952058, −8.875019101495954620778048483804, −7.81888505033064674159010229696, −6.53394558205461813495748411280, −6.11657793539456641230812565600, −5.22268833442395987856027492447, −3.85657705179064941227694016503, −3.39861364670798027083155691776, −2.49222628709902457140963542041, 0.74257828528487184574472604409, 2.14418466328312579044333223926, 3.81796117598964040640773777440, 4.46069151470838341193955347477, 5.30781989566304517144563400227, 6.05801642132280353862435209807, 7.09448227748943964790034996452, 8.366220553065565247428887335591, 8.756718683151601829722448783275, 9.696455120417960840476269819055

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