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Properties

Label 2-1734-17.16-c1-0-30
Degree $2$
Conductor $1734$
Sign $0.985 - 0.168i$
Analytic cond. $13.8460$
Root an. cond. $3.72102$
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 + 2.34i·5-s + i·6-s − 4.06i·7-s + 8-s − 9-s + 2.34i·10-s − 2.83i·11-s + i·12-s + 2.69·13-s − 4.06i·14-s − 2.34·15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 1.04i·5-s + 0.408i·6-s − 1.53i·7-s + 0.353·8-s − 0.333·9-s + 0.742i·10-s − 0.855i·11-s + 0.288i·12-s + 0.747·13-s − 1.08i·14-s − 0.606·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(13.8460\)
Root analytic conductor: \(3.72102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1734} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1734,\ (\ :1/2),\ 0.985 - 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.878621299\)
\(L(\frac12)\) \(\approx\) \(2.878621299\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
17 \( 1 \)
good5 \( 1 - 2.34iT - 5T^{2} \)
7 \( 1 + 4.06iT - 7T^{2} \)
11 \( 1 + 2.83iT - 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
19 \( 1 - 6.45T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 8.70iT - 29T^{2} \)
31 \( 1 - 0.573iT - 31T^{2} \)
37 \( 1 + 7.88iT - 37T^{2} \)
41 \( 1 - 3.43iT - 41T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 + 4.34T + 59T^{2} \)
61 \( 1 - 13.0iT - 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + 2.36iT - 73T^{2} \)
79 \( 1 - 7.37iT - 79T^{2} \)
83 \( 1 + 3.21T + 83T^{2} \)
89 \( 1 + 6.34T + 89T^{2} \)
97 \( 1 - 8.11iT - 97T^{2} \)
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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

−9.534925590484585935277091763855, −8.476305553792238201119779625131, −7.48150721186426111323556432957, −6.90371102390815241037896940554, −6.12663378907097350440881429308, −5.18452842293973563291634712307, −4.14834395811062002030626718440, −3.48048320814634986026487139184, −2.85369326912016958250159243542, −1.03031405711476029473868916461, 1.27431455063429210593239569157, 2.23059160702758785882835037043, 3.28120378972465676665758750857, 4.50049654401077749483825359269, 5.39614757443813238091562137540, 5.76354343596798606398604027459, 6.76002913902481838200802558016, 7.80014381068236511537794464217, 8.368622053937263392440490817942, 9.337728027405551728649759802170

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