Properties

Label 2-740-185.179-c0-0-0
Degree $2$
Conductor $740$
Sign $0.646 - 0.763i$
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  − 3-s + 5-s + i·7-s + i·11-s − 15-s i·21-s + 25-s + 27-s + (1 + i)31-s i·33-s + i·35-s + i·37-s + i·41-s + (−1 − i)43-s i·47-s + ⋯
L(s)  = 1  − 3-s + 5-s + i·7-s + i·11-s − 15-s i·21-s + 25-s + 27-s + (1 + i)31-s i·33-s + i·35-s + i·37-s + i·41-s + (−1 − i)43-s i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.646 - 0.763i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.646 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7803770009\)
\(L(\frac12)\) \(\approx\) \(0.7803770009\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.56862508602636943109469905733, −9.994769794085147461570745753155, −9.103515677188854980741298460547, −8.269091570514110769839769188153, −6.81138345199920516422076649275, −6.26784537709608327171777563953, −5.30374524985244088228701352809, −4.81288738167812688382663982630, −2.93856423426109703182556366949, −1.76449077401087519533179252045, 1.02748974401997445720603175672, 2.77139635544225333279026596199, 4.17011262634207891587118467124, 5.28798658406050219517083468675, 6.02213689077084962837281446081, 6.66478762581296740870360209884, 7.79575930962859574121482498576, 8.867974839052776531917143822463, 9.799286297463342101753300914211, 10.67516636882129806210701009476

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