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Properties

Label 2-1682-29.28-c1-0-67
Degree $2$
Conductor $1682$
Sign $-0.371 - 0.928i$
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 − 3i·3-s − 4-s + 3·5-s − 3·6-s − 2·7-s + i·8-s − 6·9-s − 3i·10-s − i·11-s + 3i·12-s − 3·13-s + 2i·14-s − 9i·15-s + 16-s − 4i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.73i·3-s − 0.5·4-s + 1.34·5-s − 1.22·6-s − 0.755·7-s + 0.353i·8-s − 2·9-s − 0.948i·10-s − 0.301i·11-s + 0.866i·12-s − 0.832·13-s + 0.534i·14-s − 2.32i·15-s + 0.250·16-s − 0.970i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1682\)    =    \(2 \cdot 29^{2}\)
Sign: $-0.371 - 0.928i$
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: \((2,\ 1682,\ (\ :1/2),\ -0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.167896994\)
\(L(\frac12)\) \(\approx\) \(1.167896994\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 3T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + iT - 11T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
31 \( 1 - 3iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 - 7iT - 43T^{2} \)
47 \( 1 + 11iT - 47T^{2} \)
53 \( 1 - T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 + 7iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−8.984475582426802165078203081386, −8.018388553981976775876385309011, −6.87587112949016873299982435127, −6.68014073435152539812124297049, −5.62369630595312297732320791867, −4.87616465118659609228856566843, −2.91250706337636848359381278780, −2.60971144266374629866654058573, −1.54415424273204574331276588812, −0.41981258853431780794457895063, 2.11057303024008854148953518259, 3.41691704988947794663062719915, 4.15947300507508592625729898362, 5.13188491016681541977279145476, 5.84607367904095129678844366097, 6.25288933971183422438460006396, 7.56525463191707240072684328125, 8.566060319151987108241448223752, 9.443036635129054403487352549361, 9.722069607411543082928815530656

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