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Properties

Label 2-1734-17.16-c1-0-42
Degree $2$
Conductor $1734$
Sign $-0.970 + 0.242i$
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 − 4i·5-s − i·6-s + 2i·7-s + 8-s − 9-s − 4i·10-s − i·12-s − 6·13-s + 2i·14-s − 4·15-s + 16-s − 18-s − 4·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 1.78i·5-s − 0.408i·6-s + 0.755i·7-s + 0.353·8-s − 0.333·9-s − 1.26i·10-s − 0.288i·12-s − 1.66·13-s + 0.534i·14-s − 1.03·15-s + 0.250·16-s − 0.235·18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $-0.970 + 0.242i$
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: \((2,\ 1734,\ (\ :1/2),\ -0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.621643912\)
\(L(\frac12)\) \(\approx\) \(1.621643912\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 5T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 6iT - 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 4iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
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

−8.932550138820303057236108290575, −8.083719785589114210151009144660, −7.48478081409387095763036935873, −6.24294333150010171645721185796, −5.67410378192764600999576970252, −4.70040527668591693864281822378, −4.36230703799187943702606368017, −2.63602006407950501680318947270, −1.93204096322546779542622862488, −0.43190188565700093538794961374, 2.13389107667498561038475665094, 3.03377848780362216424177989870, 3.72549137367391334076095162155, 4.63990908606823152588077389965, 5.58723895218723377670047584683, 6.55522925949597452392085973318, 7.24168772957863884556814548472, 7.62468387367011751838087182596, 9.075111765818851090833807146395, 10.07267221508283389549390052382

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