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Properties

Label 2-1734-17.16-c1-0-45
Degree $2$
Conductor $1734$
Sign $-0.896 + 0.443i$
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 − 1.26i·5-s − i·6-s − 2.94i·7-s + 8-s − 9-s − 1.26i·10-s − 3.29i·11-s − i·12-s − 3.36·13-s − 2.94i·14-s − 1.26·15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.564i·5-s − 0.408i·6-s − 1.11i·7-s + 0.353·8-s − 0.333·9-s − 0.399i·10-s − 0.994i·11-s − 0.288i·12-s − 0.932·13-s − 0.787i·14-s − 0.325·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $-0.896 + 0.443i$
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: \((2,\ 1734,\ (\ :1/2),\ -0.896 + 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.907044705\)
\(L(\frac12)\) \(\approx\) \(1.907044705\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.26iT - 5T^{2} \)
7 \( 1 + 2.94iT - 7T^{2} \)
11 \( 1 + 3.29iT - 11T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 - 0.522iT - 29T^{2} \)
31 \( 1 - 7.10iT - 31T^{2} \)
37 \( 1 + 0.792iT - 37T^{2} \)
41 \( 1 + 4.06iT - 41T^{2} \)
43 \( 1 + 0.867T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 9.98T + 53T^{2} \)
59 \( 1 - 7.31T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 - 7.38iT - 71T^{2} \)
73 \( 1 + 5.23iT - 73T^{2} \)
79 \( 1 + 10.9iT - 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 + 6.22iT - 97T^{2} \)
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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

−8.679157142243481332312729116301, −8.215869786004453522213877299382, −7.09587933249449598867511629045, −6.73102741631597530741000273187, −5.66015021827044004464235696025, −4.83973476862714870273129331137, −4.00504193724180237559654060083, −3.02414521977156926856967093405, −1.79474663568171014402348298939, −0.52186634132046378823888384189, 2.24213738908307120788235706433, 2.62197127426860465597306986502, 3.95519537529711478784641222692, 4.69998676121281520758330957366, 5.46404634046350985134862120576, 6.38780569111831657835455888311, 7.02505219458175667634803461640, 8.098086336276695250892876357811, 8.897147667461682549951519932917, 9.855491044738629783815228067534

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