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Properties

Label 2-1740-1740.347-c0-0-4
Degree $2$
Conductor $1740$
Sign $0.229 - 0.973i$
Analytic cond. $0.868373$
Root an. cond. $0.931865$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.965 + 0.258i)3-s − 1.00i·4-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.500i)6-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.258 − 0.965i)10-s − 0.517i·11-s + (0.258 − 0.965i)12-s + (1.36 − 1.36i)13-s + (−0.707 + 0.707i)15-s − 1.00·16-s + (−0.965 + 0.258i)18-s + 1.41i·19-s + (0.866 + 0.500i)20-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.965 + 0.258i)3-s − 1.00i·4-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.500i)6-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.258 − 0.965i)10-s − 0.517i·11-s + (0.258 − 0.965i)12-s + (1.36 − 1.36i)13-s + (−0.707 + 0.707i)15-s − 1.00·16-s + (−0.965 + 0.258i)18-s + 1.41i·19-s + (0.866 + 0.500i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(0.868373\)
Root analytic conductor: \(0.931865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :0),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100118759\)
\(L(\frac12)\) \(\approx\) \(1.100118759\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + 0.517iT - T^{2} \)
13 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - iT^{2} \)
31 \( 1 - 0.517T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.93iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.653307023861136877386235540679, −8.534532384821488478461238177963, −8.090865100904366959036528354496, −7.72118562994037220679854086818, −6.50589032390008631793884529893, −6.01709356966253001520038587948, −4.76472735479220015593748014203, −3.58876636408243406882430936914, −2.93739802901532500929982336882, −1.42102008804650074646400752708, 1.17654470838793353186971995529, 2.08749309471942615241661129992, 3.28295562416785701597970150050, 4.12822176593897267908685090859, 4.76796261795818881804319049977, 6.62177527530703877729981099986, 7.11665935546558348038615293504, 8.221336948596840343416919075697, 8.583414417162089165668108904244, 9.199272734735406085247546194141

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