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Properties

Label 2-740-740.347-c0-0-0
Degree $2$
Conductor $740$
Sign $0.166 - 0.986i$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s + 0.999i·8-s + (−0.866 − 0.5i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.866 + 0.499i)20-s − 25-s + (1.36 − 1.36i)29-s + (−0.866 + 0.499i)32-s + (0.866 − 0.499i)34-s − 0.999i·36-s + (−0.866 + 0.5i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s + 0.999i·8-s + (−0.866 − 0.5i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.866 + 0.499i)20-s − 25-s + (1.36 − 1.36i)29-s + (−0.866 + 0.499i)32-s + (0.866 − 0.499i)34-s − 0.999i·36-s + (−0.866 + 0.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.166 - 0.986i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.166 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.464052464\)
\(L(\frac12)\) \(\approx\) \(1.464052464\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 - iT \)
37 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
31 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + 1.73T + T^{2} \)
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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

−10.99084945834413878145983543302, −10.02913424666341678169909155311, −8.904932170696850592005342927386, −7.925050925981699548378780947405, −7.13405408471555076256760807172, −6.27187084497638680750546371118, −5.61800805877455403063854783559, −4.38192199185416200915025067144, −3.24791354945668227586490716986, −2.55604529841625931071427480001, 1.44415463942121210603204209397, 2.80144763278416741010262044482, 3.96896136100766709259412529670, 5.00090472267369864288036427568, 5.60499459741940231579639292109, 6.59045170453058589149057285217, 7.921243420314088278954123553002, 8.706323272906228129238647344075, 9.674016081551268435560383385089, 10.59878287025958678488902314947

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