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Properties

Label 2-867-17.16-c1-0-15
Degree $2$
Conductor $867$
Sign $-0.371 - 0.928i$
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  + 1.84·2-s + i·3-s + 1.41·4-s + 1.61i·5-s + 1.84i·6-s + 0.152i·7-s − 1.08·8-s − 9-s + 2.98i·10-s + 5.02i·11-s + 1.41i·12-s − 3.94·13-s + 0.281i·14-s − 1.61·15-s − 4.82·16-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.577i·3-s + 0.707·4-s + 0.721i·5-s + 0.754i·6-s + 0.0575i·7-s − 0.382·8-s − 0.333·9-s + 0.942i·10-s + 1.51i·11-s + 0.408i·12-s − 1.09·13-s + 0.0751i·14-s − 0.416·15-s − 1.20·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.371 - 0.928i$
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: \((2,\ 867,\ (\ :1/2),\ -0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37566 + 2.03158i\)
\(L(\frac12)\) \(\approx\) \(1.37566 + 2.03158i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.84T + 2T^{2} \)
5 \( 1 - 1.61iT - 5T^{2} \)
7 \( 1 - 0.152iT - 7T^{2} \)
11 \( 1 - 5.02iT - 11T^{2} \)
13 \( 1 + 3.94T + 13T^{2} \)
19 \( 1 - 6.57T + 19T^{2} \)
23 \( 1 - 3.44iT - 23T^{2} \)
29 \( 1 - 2.24iT - 29T^{2} \)
31 \( 1 - 2.57iT - 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 + 0.276iT - 41T^{2} \)
43 \( 1 + 6.34T + 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 - 2.12T + 53T^{2} \)
59 \( 1 + 1.32T + 59T^{2} \)
61 \( 1 - 8.27iT - 61T^{2} \)
67 \( 1 - 7.10T + 67T^{2} \)
71 \( 1 - 6.54iT - 71T^{2} \)
73 \( 1 + 8.32iT - 73T^{2} \)
79 \( 1 - 0.532iT - 79T^{2} \)
83 \( 1 + 2.20T + 83T^{2} \)
89 \( 1 + 7.64T + 89T^{2} \)
97 \( 1 + 3.60iT - 97T^{2} \)
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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

−10.43621669478678265458712000689, −9.705980875462427071402479646994, −9.008336882474207510176432203782, −7.33667100834847610527301188265, −7.06084782803166385291227190267, −5.66013407557939945211192259898, −5.05717548311848312967079720464, −4.20005996886951056415800298007, −3.22614756992582128076243598704, −2.32255779525893652404522640468, 0.77022897411635701593274219856, 2.59971387892743302803092863763, 3.46303924148618432821873452135, 4.70379076520893989101159289559, 5.36576855130608592137136494567, 6.14899372020190799575838444168, 7.10763702502036004115457803438, 8.182972674439030260971790545244, 8.913329759065641867262373274005, 9.875291840885778618985684926942

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