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Properties

Label 2-760-8.5-c1-0-66
Degree $2$
Conductor $760$
Sign $-0.813 + 0.581i$
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  + (0.943 + 1.05i)2-s − 2.94i·3-s + (−0.219 + 1.98i)4-s − i·5-s + (3.10 − 2.77i)6-s − 3.06·7-s + (−2.30 + 1.64i)8-s − 5.66·9-s + (1.05 − 0.943i)10-s + 1.87i·11-s + (5.85 + 0.647i)12-s − 5.87i·13-s + (−2.89 − 3.23i)14-s − 2.94·15-s + (−3.90 − 0.874i)16-s − 5.65·17-s + ⋯
L(s)  = 1  + (0.667 + 0.744i)2-s − 1.69i·3-s + (−0.109 + 0.993i)4-s − 0.447i·5-s + (1.26 − 1.13i)6-s − 1.15·7-s + (−0.813 + 0.581i)8-s − 1.88·9-s + (0.333 − 0.298i)10-s + 0.565i·11-s + (1.68 + 0.186i)12-s − 1.62i·13-s + (−0.773 − 0.863i)14-s − 0.760·15-s + (−0.975 − 0.218i)16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.813 + 0.581i$
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: \((2,\ 760,\ (\ :1/2),\ -0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266416 - 0.831543i\)
\(L(\frac12)\) \(\approx\) \(0.266416 - 0.831543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.943 - 1.05i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 + 2.94iT - 3T^{2} \)
7 \( 1 + 3.06T + 7T^{2} \)
11 \( 1 - 1.87iT - 11T^{2} \)
13 \( 1 + 5.87iT - 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
23 \( 1 + 1.68T + 23T^{2} \)
29 \( 1 - 0.0474iT - 29T^{2} \)
31 \( 1 - 6.54T + 31T^{2} \)
37 \( 1 + 7.10iT - 37T^{2} \)
41 \( 1 + 3.78T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + 0.994T + 47T^{2} \)
53 \( 1 + 9.56iT - 53T^{2} \)
59 \( 1 - 6.49iT - 59T^{2} \)
61 \( 1 + 4.53iT - 61T^{2} \)
67 \( 1 + 7.53iT - 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 9.77T + 73T^{2} \)
79 \( 1 + 4.30T + 79T^{2} \)
83 \( 1 + 1.69iT - 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 19.5T + 97T^{2} \)
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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

−9.799505672606488275856558106356, −8.690898605896608628947417761877, −8.006299054416342862800246270639, −7.22959404724489628846246411240, −6.46550664872691264218257106460, −5.92014920622246669533260180262, −4.77116472397237875054929156528, −3.27912316873237443122840194049, −2.30799946774774125332641647956, −0.32754011755540863886630435335, 2.43834107051336527103029653240, 3.45434308927518249824319605762, 4.11293478601103593802681041317, 4.91631991795014403742372562106, 6.16435272454536683680490405697, 6.66678010705679902984250007872, 8.763221364626542169346843597002, 9.254083604028907488334694629920, 10.07140075257083324629271254318, 10.52748383435861255488127803762

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