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Properties

Label 2-2890-17.16-c1-0-47
Degree $2$
Conductor $2890$
Sign $0.242 + 0.970i$
Analytic cond. $23.0767$
Root an. cond. $4.80382$
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-s − i·3-s + 4-s + i·5-s + i·6-s − i·7-s − 8-s + 2·9-s − i·10-s − i·12-s − 4·13-s + i·14-s + 15-s + 16-s − 2·18-s + 4·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 0.447i·5-s + 0.408i·6-s − 0.377i·7-s − 0.353·8-s + 0.666·9-s − 0.316i·10-s − 0.288i·12-s − 1.10·13-s + 0.267i·14-s + 0.258·15-s + 0.250·16-s − 0.471·18-s + 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2890\)    =    \(2 \cdot 5 \cdot 17^{2}\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(23.0767\)
Root analytic conductor: \(4.80382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2890} (2311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2890,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.186624382\)
\(L(\frac12)\) \(\approx\) \(1.186624382\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - iT \)
17 \( 1 \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 3iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.483414034683864378188392433378, −7.56277865196401462108288061444, −7.38084310022428812238533166824, −6.63362447042921678271053809634, −5.75984933580288816085016021861, −4.72138404553947619026225792246, −3.69045354860122262526671911965, −2.61725706645985684844468860646, −1.74662292402365013999281585183, −0.56778804928950598091847590753, 1.02832473081322840057854112033, 2.19841278530057643720443064680, 3.21391623184740625602208576996, 4.29881814731718880686675369577, 5.06824685127709190899003337497, 5.75632384588285687564403170512, 7.04020430473920827106931757285, 7.31000265140466359713524703877, 8.425544194421182110423768369970, 8.938097994529038014501110173720

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