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Properties

Label 2-760-5.4-c1-0-15
Degree $2$
Conductor $760$
Sign $0.316 + 0.948i$
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  − 0.414i·3-s + (−2.12 + 0.707i)5-s − 0.414i·7-s + 2.82·9-s − 1.41·11-s − 3.82i·13-s + (0.292 + 0.878i)15-s − i·17-s + 19-s − 0.171·21-s − 3.24i·23-s + (3.99 − 3i)25-s − 2.41i·27-s + 1.82·29-s + 0.585·31-s + ⋯
L(s)  = 1  − 0.239i·3-s + (−0.948 + 0.316i)5-s − 0.156i·7-s + 0.942·9-s − 0.426·11-s − 1.06i·13-s + (0.0756 + 0.226i)15-s − 0.242i·17-s + 0.229·19-s − 0.0374·21-s − 0.676i·23-s + (0.799 − 0.600i)25-s − 0.464i·27-s + 0.339·29-s + 0.105·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.960873 - 0.692558i\)
\(L(\frac12)\) \(\approx\) \(0.960873 - 0.692558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.12 - 0.707i)T \)
19 \( 1 - T \)
good3 \( 1 + 0.414iT - 3T^{2} \)
7 \( 1 + 0.414iT - 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 3.82iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
23 \( 1 + 3.24iT - 23T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 - 0.585T + 31T^{2} \)
37 \( 1 + 10.8iT - 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 6.24iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 3.82iT - 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 0.585T + 61T^{2} \)
67 \( 1 + 8.07iT - 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 8.17iT - 73T^{2} \)
79 \( 1 + 4.82T + 79T^{2} \)
83 \( 1 - 14.4iT - 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 - 3.65iT - 97T^{2} \)
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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

−10.53593484341264417916507489587, −9.309944443354267417584833079377, −8.230411374971702330389934704258, −7.52484370215071357498131296285, −6.95903627960470791402390147268, −5.73070765141885938716729093325, −4.58791245096905576973691042751, −3.68108591139365277382034526585, −2.50766382639485264894953352878, −0.66784810928231549694847414130, 1.42128487423258976484156251315, 3.10069969980703888622577382641, 4.26823320588900387482720975867, 4.78275441402741845589119708664, 6.15065904734731538771193300256, 7.21294797274446815649898991160, 7.84814245473558443850083954746, 8.853222600977272364494905173075, 9.605734905221495732950580697189, 10.50631839468295420561996499722

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