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Properties

Label 2-1740-1740.1043-c0-0-4
Degree $2$
Conductor $1740$
Sign $-0.957 - 0.287i$
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 + (−0.258 + 0.965i)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 + (−0.965 − 0.258i)12-s + (1.36 + 1.36i)13-s + (−0.965 + 0.258i)15-s − 1.00·16-s + (−0.258 − 0.965i)18-s − 1.41i·19-s + (−0.866 + 0.500i)20-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.258 + 0.965i)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 + (−0.965 − 0.258i)12-s + (1.36 + 1.36i)13-s + (−0.965 + 0.258i)15-s − 1.00·16-s + (−0.258 − 0.965i)18-s − 1.41i·19-s + (−0.866 + 0.500i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1740\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 29\)
Sign: $-0.957 - 0.287i$
Analytic conductor: \(0.868373\)
Root analytic conductor: \(0.931865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1740} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1740,\ (\ :0),\ -0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.579968784\)
\(L(\frac12)\) \(\approx\) \(1.579968784\)
\(L(1)\) 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 \( 1 + (0.258 - 0.965i)T \)
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

−9.759771920783640878196891757360, −8.936806609429264896536469107170, −8.481447794668702494787337781650, −7.06588769674963873759200157601, −6.56617722475777760984529329007, −5.82494869363334984069385051649, −5.07910840808459341985366067971, −3.99707675676671452261006800081, −3.46262817259230496020142211411, −2.34755244184361478723333636946, 1.07581276732276529808522626315, 1.83031353901386795734633010848, 3.03322902411396588563828602995, 4.10270677474323177260831299459, 5.23902436820434293308711956847, 5.82367980337799468959254648432, 6.32457773736774503365693445836, 7.61435597535467054227279573789, 8.359179424698103105905091874919, 9.173179984170486519052554524466

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