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Properties

Label 2-760-8.5-c1-0-28
Degree $2$
Conductor $760$
Sign $0.390 + 0.920i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (−0.188 − 1.40i)2-s − 0.0766i·3-s + (−1.92 + 0.528i)4-s − i·5-s + (−0.107 + 0.0144i)6-s + 1.59·7-s + (1.10 + 2.60i)8-s + 2.99·9-s + (−1.40 + 0.188i)10-s − 0.418i·11-s + (0.0404 + 0.147i)12-s + 6.12i·13-s + (−0.301 − 2.24i)14-s − 0.0766·15-s + (3.44 − 2.03i)16-s + 2.23·17-s + ⋯
L(s)  = 1  + (−0.133 − 0.991i)2-s − 0.0442i·3-s + (−0.964 + 0.264i)4-s − 0.447i·5-s + (−0.0438 + 0.00589i)6-s + 0.604·7-s + (0.390 + 0.920i)8-s + 0.998·9-s + (−0.443 + 0.0595i)10-s − 0.126i·11-s + (0.0116 + 0.0426i)12-s + 1.69i·13-s + (−0.0805 − 0.599i)14-s − 0.0197·15-s + (0.860 − 0.509i)16-s + 0.541·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26753 - 0.839572i\)
\(L(\frac12)\) \(\approx\) \(1.26753 - 0.839572i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.188 + 1.40i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 + 0.0766iT - 3T^{2} \)
7 \( 1 - 1.59T + 7T^{2} \)
11 \( 1 + 0.418iT - 11T^{2} \)
13 \( 1 - 6.12iT - 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
23 \( 1 - 4.81T + 23T^{2} \)
29 \( 1 - 4.65iT - 29T^{2} \)
31 \( 1 - 1.44T + 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 0.923T + 41T^{2} \)
43 \( 1 + 4.94iT - 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 + 8.15iT - 59T^{2} \)
61 \( 1 - 10.8iT - 61T^{2} \)
67 \( 1 + 5.14iT - 67T^{2} \)
71 \( 1 - 4.56T + 71T^{2} \)
73 \( 1 - 3.75T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 + 0.561iT - 83T^{2} \)
89 \( 1 + 1.07T + 89T^{2} \)
97 \( 1 + 6.13T + 97T^{2} \)
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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

−10.26249641557970528316262303602, −9.241265245935919799215138511524, −8.882850869974599540661535988623, −7.71709210843888434402102367554, −6.87358408475116798696811988378, −5.32337348041074503923634020213, −4.52232999594909175598622500035, −3.73560145401401573548446549224, −2.14691569334743990670934507387, −1.18731237410922140904475646398, 1.13946885672014736048374332977, 3.11094808353916089716517270157, 4.36370677348949516020779662473, 5.22943935392109102302121949615, 6.13955092439360437093662513774, 7.17978666109328950513225220602, 7.80170359858889633334303662903, 8.486952849976548941558847730301, 9.775789439426873928589550733600, 10.18636339451533394058253894961

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