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Properties

Label 2-1682-29.28-c1-0-53
Degree $2$
Conductor $1682$
Sign $0.233 + 0.972i$
Analytic cond. $13.4308$
Root an. cond. $3.66481$
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  − i·2-s − 0.710i·3-s − 4-s + 3.33·5-s − 0.710·6-s + 4.40·7-s + i·8-s + 2.49·9-s − 3.33i·10-s + 2.39i·11-s + 0.710i·12-s − 5.40·13-s − 4.40i·14-s − 2.36i·15-s + 16-s − 3.10i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.409i·3-s − 0.5·4-s + 1.49·5-s − 0.289·6-s + 1.66·7-s + 0.353i·8-s + 0.831·9-s − 1.05i·10-s + 0.723i·11-s + 0.204i·12-s − 1.49·13-s − 1.17i·14-s − 0.611i·15-s + 0.250·16-s − 0.753i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1682\)    =    \(2 \cdot 29^{2}\)
Sign: $0.233 + 0.972i$
Analytic conductor: \(13.4308\)
Root analytic conductor: \(3.66481\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1682} (1681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1682,\ (\ :1/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.647831798\)
\(L(\frac12)\) \(\approx\) \(2.647831798\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
29 \( 1 \)
good3 \( 1 + 0.710iT - 3T^{2} \)
5 \( 1 - 3.33T + 5T^{2} \)
7 \( 1 - 4.40T + 7T^{2} \)
11 \( 1 - 2.39iT - 11T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 + 3.55iT - 19T^{2} \)
23 \( 1 + 0.408T + 23T^{2} \)
31 \( 1 - 1.23iT - 31T^{2} \)
37 \( 1 - 6.21iT - 37T^{2} \)
41 \( 1 + 3.97iT - 41T^{2} \)
43 \( 1 + 6.00iT - 43T^{2} \)
47 \( 1 - 8.56iT - 47T^{2} \)
53 \( 1 - 5.49T + 53T^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + 8.27T + 67T^{2} \)
71 \( 1 + 4.70T + 71T^{2} \)
73 \( 1 + 2.91iT - 73T^{2} \)
79 \( 1 + 5.81iT - 79T^{2} \)
83 \( 1 + 7.60T + 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 + 16.9iT - 97T^{2} \)
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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

−9.392242853942785505404988898178, −8.573876872574147112108976053249, −7.41441452469863887903655251969, −7.07145298898570914768697585827, −5.68990853561515007527602756686, −4.84318988149459090995843144720, −4.52315268520743071957567093297, −2.56756896218501661480904895545, −2.04417911003482456662543305314, −1.21131686720096201216714704877, 1.42320691757046135135956033633, 2.27627223719113904449528713559, 3.93889840920705536850731612216, 4.86837399489284619635976424497, 5.38181724819280566264068352464, 6.14363949678759785520239893663, 7.18518700405543176701410890370, 7.925132841357985671795056841615, 8.681714745600964449079617941820, 9.560845736556520019438282604609

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