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Properties

Label 2-1692-141.140-c1-0-6
Degree $2$
Conductor $1692$
Sign $0.458 - 0.888i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1692 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1692\)    =    \(2^{2} \cdot 3^{2} \cdot 47\)
Sign: $0.458 - 0.888i$
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: \((2,\ 1692,\ (\ :1/2),\ 0.458 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.806671509\)
\(L(\frac12)\) \(\approx\) \(1.806671509\)
\(L(\frac{3}{2})\) 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

−9.476358771008406799453844249564, −9.096256952307581885537569923861, −7.939063381797779392630949440706, −6.61864850272577172932347991762, −6.38200021557762614444148318446, −5.76726846674842752663880422458, −4.34776323235105496590676224234, −3.61700893912832651302783078263, −2.35775258382544530389815863792, −1.41784741352207112416562961770, 0.70431393938166138334697569633, 2.19123121770196474903057373524, 3.10465339877805147736590339552, 4.05634265587161508782374987204, 5.44140589096594047746874515888, 5.92678893574530987503794936695, 6.69271903974795552028657469692, 7.42931845081605413378598386277, 8.676448598483109970549735232980, 9.454956049606914269257220557950

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