Properties

Label 2-1692-141.140-c1-0-3
Degree $2$
Conductor $1692$
Sign $0.990 - 0.136i$
Analytic cond. $13.5106$
Root an. cond. $3.67568$
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  − 2.84·5-s − 3.53·7-s − 4.06·11-s − 4.24i·13-s + 5.67i·17-s + 0.496i·19-s + 3.35·23-s + 3.10·25-s + 3.85·29-s − 0.715i·31-s + 10.0·35-s + 10.1·37-s − 0.201·41-s − 9.49i·43-s + (−3.15 + 6.08i)47-s + ⋯
L(s)  = 1  − 1.27·5-s − 1.33·7-s − 1.22·11-s − 1.17i·13-s + 1.37i·17-s + 0.113i·19-s + 0.700·23-s + 0.621·25-s + 0.716·29-s − 0.128i·31-s + 1.70·35-s + 1.66·37-s − 0.0315·41-s − 1.44i·43-s + (−0.460 + 0.887i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1692\)    =    \(2^{2} \cdot 3^{2} \cdot 47\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(13.5106\)
Root analytic conductor: \(3.67568\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1692} (845, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1692,\ (\ :1/2),\ 0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7195226528\)
\(L(\frac12)\) \(\approx\) \(0.7195226528\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
47 \( 1 + (3.15 - 6.08i)T \)
good5 \( 1 + 2.84T + 5T^{2} \)
7 \( 1 + 3.53T + 7T^{2} \)
11 \( 1 + 4.06T + 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 5.67iT - 17T^{2} \)
19 \( 1 - 0.496iT - 19T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 + 0.715iT - 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 + 0.201T + 41T^{2} \)
43 \( 1 + 9.49iT - 43T^{2} \)
53 \( 1 - 3.66iT - 53T^{2} \)
59 \( 1 + 9.26iT - 59T^{2} \)
61 \( 1 - 8.60T + 61T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 - 4.39iT - 71T^{2} \)
73 \( 1 + 7.98iT - 73T^{2} \)
79 \( 1 - 5.53T + 79T^{2} \)
83 \( 1 + 2.85iT - 83T^{2} \)
89 \( 1 - 7.77iT - 89T^{2} \)
97 \( 1 + 4.96T + 97T^{2} \)
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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

−9.397011960172762334379260072651, −8.248670668574440131243122033828, −7.974410176777568417323203761512, −7.05878865382162661581496368103, −6.15377806308008670619016734035, −5.32852127734006151118489228873, −4.17814283112447701339191146672, −3.37715255467352710987925375862, −2.66409202908218964719721841817, −0.59140789104505726730238876397, 0.52467597757697256279630802032, 2.60520127249378066443753599169, 3.27088279798742499622900846440, 4.32000790320876233315058785862, 5.05594669482318339534588281730, 6.30529857592330041730086808562, 7.02089902978941212302469609383, 7.64601322660076155909881318913, 8.497958105283252646991820372753, 9.420702068098427412839583861255

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